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3338Boundary Conditions MatterWhy do boundary conditions qualitatively matter for heat diffusion on a finite interval?数学中等essay未尝试面试订阅3339Diffusion as Local AveragingExplain in words why the heat equation can be viewed as repeated local averaging.数学中等essay未尝试面试订阅3340Why Diffusion Forgets Fine Initial DetailsWhy does long-time heat evolution remember coarse mass distribution more than fine initial details?数学中等essay未尝试面试订阅3464Why Label Leakage Makes Mutual Information Look Too GoodWhy can label leakage or look-ahead information make estimated mutual information appear enormous in-sample but vanish out of sample?数学中等essay未尝试面试订阅3466Capacity of a Ternary Test With Uneven Follow-UpsA first test has three outcomes. If outcome A occurs, you may ask two binary follow-up questions; if outcome B occurs, you may ask one ternary follow-up; if outcome C occurs, the process must stop. What is the maximum number of equally likely states you can distinguish?数学简单derivation未尝试面试订阅3467Capacity of a Binary Gate With Different Menus on Each SideA first yes/no question is asked. On the yes branch, you may run one four-way diagnostic. On the no branch, you may run two further yes/no diagnostics. What is the maximum number of states that can be distinguished?数学简单derivation未尝试面试订阅3468Capacity of a Four-Way Triage With Partial Follow-UpA first test has four outcomes. On the first two branches you may ask one binary follow-up each; on the last two branches you must stop immediately. What is the maximum number of states you can distinguish?数学简单derivation未尝试面试订阅3469Capacity of a Ternary Gate With a Five-Way EscalationA first test has outcomes left, middle, right. If left occurs, you may run one five-way escalation test. If middle or right occurs, you may run one binary follow-up on that branch. How many states can be distinguished in total?数学简单derivation未尝试面试订阅3470Capacity of a Binary Gate With Deep and Shallow BranchesA first yes/no question is asked. If yes, you may then ask one ternary question followed by one binary question. If no, you may ask only one ternary question. What is the maximum number of distinguishable states?数学简单derivation未尝试面试订阅3476Entropy Lower Bound for a 1/2, 1/4, 1/8, 1/8 PriorA hidden state has prior probabilities [0.5, 0.25, 0.125, 0.125]. What is the Shannon lower bound, in yes/no questions on average, for any adaptive identification strategy?数学中等derivation未尝试面试订阅3477Entropy Lower Bound for a 1/2, 1/4, 1/4 PriorA hidden state has prior probabilities [0.5, 0.25, 0.25]. What is the Shannon lower bound, in yes/no questions on average, for any adaptive identification strategy?数学中等derivation未尝试面试订阅3478Entropy Lower Bound for a 1/4, 1/4, 1/4, 1/8, 1/8 PriorA hidden state has prior probabilities [0.25, 0.25, 0.25, 0.125, 0.125]. What is the Shannon lower bound, in yes/no questions on average, for any adaptive identification strategy?数学中等derivation未尝试面试订阅3480Entropy Lower Bound for a Six-State Dyadic PriorA hidden state has prior probabilities [0.25, 0.25, 0.125, 0.125, 0.125, 0.125]. What is the Shannon lower bound, in yes/no questions on average, for any adaptive identification strategy?数学中等derivation未尝试面试订阅3486Why Adaptive Testing Beats Static MenusWhy can an adaptive questioning strategy distinguish more states than a non-adaptive test battery with the same primitive tests?数学中等essay未尝试面试订阅3487Why Entropy Is a Lower Bound, Not Always an Exact CountWhy does the Shannon entropy of the prior give only a lower bound on expected yes/no questions rather than always the exact optimum?数学中等essay未尝试面试订阅3488Why a None State Changes the Coding ProblemIn one-defective search problems, why does adding the possibility of no defect change the counting argument in a substantive way?数学中等essay未尝试面试订阅3489Why Prior Skew Should Change the Query TreeWhy should the optimal query tree usually isolate high-probability states earlier than low-probability states?数学中等essay未尝试面试订阅3490Why Branch Constraints Can Matter More Than Alphabet SizeWhy can two testing systems with the same maximum branch count per question have very different identification power once branch-specific follow-up rules are imposed?数学中等essay未尝试面试订阅3496Variance of 2W_1 - W_4What is Var(2W 1 - W 4)?随机过程简单derivation未尝试面试订阅3497Covariance of Two Overlapping IncrementsWhat is Cov(W 4 - W 1, W 6 - W 3)?随机过程简单derivation未尝试面试订阅