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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3410Second-Order Forward Stencil at a BoundaryUse the second-order forward stencil (-3f(0)+4f(h)-f(2h))/(2h) with h=0.1, f(0)=1, f(0.1)=1.24, and f(0.2)=1.52. What derivative estimate do you get at 0?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3416Entropy of a One-Heavy Rest-Uniform Source 1A discrete source has one special symbol of probability 0.5 and 4 other symbols sharing the remaining probability uniformly. What is the entropy in bits?数学简单derivation未尝试面试订阅3421Entropy Gain From Splitting a Quarter-Mass SymbolA source contains a symbol of probability 1/4. If you split that symbol into four equally likely sublabels while leaving everything else unchanged, by how many bits does the entropy increase?数学简单derivation未尝试面试订阅3422Entropy Gain From Splitting a 30% State Into ThreeA source has a single state of probability 0.3. If that state is refined into three equally likely sublabels, how many bits of entropy are added?数学简单derivation未尝试面试订阅3423Entropy Loss From Merging Two Equal 1/8 StatesIf two symbols each of probability 1/8 are merged into a single symbol while everything else stays unchanged, by how many bits does the entropy decrease?数学简单derivation未尝试面试订阅3424Entropy Loss From Merging a 20% State and a 30% StateTwo states of probabilities 0.2 and 0.3 are merged into a single 0.5-mass state. By how many bits does the entropy fall?数学简单derivation未尝试面试订阅3425Entropy Gain From Splitting a 12% State Into EightA symbol of probability 0.12 is split into eight equally likely sublabels. How many bits of entropy are added?数学简单derivation未尝试面试订阅3426Remaining Entropy After Revealing a 3-4-5 Bucket LabelA source is uniform over 12 states. You reveal which bucket the state lies in, where the bucket sizes are 3, 4, and 5. What is the remaining entropy H(X|bucket)?数学简单derivation未尝试面试订阅3427Remaining Entropy After Revealing a 1-3-4-8 PartitionA source is uniform over 16 states. A side signal reveals whether the realized state lies in a block of size 1, 3, 4, or 8. What is H(X|signal)?数学简单derivation未尝试面试订阅3428Remaining Entropy After a 4-vs-6 Split SignalA source is uniform over 10 states. You reveal whether the outcome lies in the first 4 states or in the last 6 states. What is the remaining entropy?数学简单derivation未尝试面试订阅3429Entropy of a Three-Regime Disjoint Alphabet SourceA regime label takes values A, B, C with probabilities 0.2, 0.5, 0.3. Conditional on A the source is deterministic, conditional on B it is uniform over 4 symbols, and conditional on C it is uniform over 2 symbols. Assuming the symbol sets are disjoint across regimes, what is the total source entropy?数学简单derivation未尝试面试订阅3430Entropy of a Fair 3-vs-5 Regime MixtureA fair regime bit first chooses source A or B. Source A is uniform over 3 symbols and source B is uniform over 5 symbols, with disjoint alphabets. What is the total entropy of the emitted symbol?数学简单derivation未尝试面试订阅3431Fixed-Length Gap for a Uniform 10-State SourceA source is uniform over 10 symbols. If you insist on a fixed-length binary code, what is the gap between the required fixed length and the entropy?数学中等derivation未尝试面试订阅