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3286Why Sign Flips of PCs Do Not MatterWhy are principal components only identified up to sign?数学中等essay未尝试面试订阅3287Why PCA Can Miss a Predictive DirectionWhy can a low-variance direction be more useful for prediction than the first principal component?数学简单essay未尝试面试订阅3288Why Whitening Can Hurt Economic InterpretabilityWhy can whitening be numerically useful while simultaneously making a factor model harder to interpret economically?数学中等essay未尝试面试订阅3289Why Truncated SVD Is a Structured DenoiserWhy does low-rank truncation often denoise a matrix instead of merely compressing it?数学中等essay未尝试面试订阅3290Why Similar Spectra Can Hide Different SubspacesWhy do similar explained-variance ratios not guarantee that two datasets have learned the same principal directions?数学中等essay未尝试面试订阅3291Exponential Decay After Two Time UnitsSolve x'(t)=-3x(t) with x(0)=12, and evaluate x(2).数学简单derivation未尝试面试订阅3293Cooling Model Over Half a UnitSolve x'(t)=-2x(t) with x(0)=10, and evaluate x( 1 2 ).数学简单derivation未尝试面试订阅3296Linear ODE with Constant ForcingSolve x'(t)+2x(t)=6 with x(0)=1, and simplify the solution.数学中等derivation未尝试面试订阅3301Logistic Growth Carrying CapacityConsider the logistic ODE x'(t)=1x(t) (1- x(t) 10 ) with initial value x(0)=2. What is the long-run limit of x(t) as t ?数学简单derivation未尝试面试订阅3306Simple Harmonic Oscillator SolutionSolve y''+4y=0 with y(0)=1 and y'(0)=0.数学中等derivation未尝试面试订阅3311Stable Equilibrium of x(1-x)Consider x'(t)=x(1-x). Which equilibrium is stable?数学中等derivation未尝试面试订阅3315Stable Root in a Shifted Logistic FlowConsider x'(t)=(x-1)(3-x). Which equilibrium is stable?数学中等derivation未尝试面试订阅3316Decay of the First Sine ModeFor the heat equation u t=1u xx on 0<x< with zero boundary conditions, the initial condition is u(x,0)=\sin(1x). What is u(x,t)?数学中等derivation未尝试面试订阅3321Two-Mode Initial ConditionFor u t=1u xx on 0<x< with zero boundary conditions and initial condition u(x,0)=1\sin(1x)+2\sin(2x), write u(x,t).数学中等derivation未尝试面试订阅3323Three-Mode Heat ProfileFor u t=1u xx on 0<x< with zero boundary conditions and initial condition u(x,0)=2\sin(1x)+1\sin(2x)+1\sin(3x), write u(x,t).数学中等derivation未尝试面试订阅3325Alternating Signs in Initial HeatFor u t=1u xx on 0<x< with zero boundary conditions and initial condition u(x,0)=1\sin(1x)-2\sin(2x), write u(x,t).数学中等derivation未尝试面试订阅3331Which Fourier Mode Decays More Slowly?Under the heat equation on 0<x< , which survives longer: the mode \sin x or the mode \sin(3x)?数学中等derivation未尝试面试订阅3332Relative Decay of the Second Mode to the FirstIf two heat modes start with the same amplitude, \sin x and \sin(2x), what multiplicative factor relates the amplitude of the second to the first after time t?数学中等derivation未尝试面试订阅3333Same Mode, Larger Diffusion CoefficientFor the same initial mode \sin(2x), which system cools faster: \kappa=1 or \kappa=3?数学中等derivation未尝试面试订阅3334Long-Run Dominant Mode in a Two-Mode MixtureIf the initial condition is u(x,0)=5\sin x+\sin(4x) with zero boundary conditions, which mode dominates at long horizons?数学中等derivation未尝试面试订阅