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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3876Futures versus Forward under Rate Correlation 1A long futures account is currently margined at balance 6200 with current settlement 102, maintenance margin 5800, and multiplier 100. What is the lowest next settlement price that still avoids a margin call?金融与交易中等derivation未尝试面试订阅3877Futures versus Forward under Rate Correlation 2A short futures account is currently margined at balance 7100 with current settlement 75, maintenance margin 6500, and multiplier 80. What is the highest next settlement price that still avoids a margin call?金融与交易中等derivation未尝试面试订阅4071Growth Retention under Fractional Kelly 1Under the quadratic Kelly scaling rule, the expected growth at fraction c of full Kelly is proportional to 2c-c 2 relative to full Kelly. What fraction of full-Kelly growth is retained at c=0.5?金融与交易中等derivation未尝试面试订阅4076Edge Estimate Is NoisyYour estimated edge is highly uncertain and may be overstated. Which is usually more prudent: full Kelly or half Kelly?金融与交易中等derivation未尝试面试订阅4077Drawdown Constraint Is BindingA mandate cares heavily about avoiding large drawdowns even if that sacrifices some long-run growth. Does that point toward full Kelly or a fractional Kelly rule?金融与交易中等derivation未尝试面试订阅4078Very Stable Edge and Deep Capital BaseIf the edge estimate is unusually stable, the capital base is deep, and drawdown tolerance is high, which becomes easier to justify: full Kelly or quarter Kelly?金融与交易中等derivation未尝试面试订阅4079Same Edge but Higher VolatilityTwo strategies have the same estimated edge, but one has much higher realized volatility and fatter drawdowns. Which should be shrunk more aggressively from full Kelly?金融与交易中等derivation未尝试面试订阅4080Unknown Correlation Across BetsIf you suspect your supposedly independent bets may actually become correlated under stress, should that make you more comfortable with full Kelly or less?金融与交易中等derivation未尝试面试订阅4083Fractional Kelly Portfolio Leverage 3A strategy has full-Kelly expected log-growth 12% per year. Under 40% Kelly and the quadratic approximation, what expected log-growth remains?金融与交易简单derivation未尝试面试订阅4084Fractional Kelly Portfolio Leverage 4To retain 75% of full-Kelly growth under the quadratic approximation with lambda in [0,1], what Kelly fraction is needed?金融与交易简单derivation未尝试面试订阅4086Why Most Real Desks Use Fractional KellyWhy do real trading desks and portfolio managers so often use fractional Kelly instead of full Kelly?金融与交易中等essay未尝试面试订阅4087Why Half Kelly Keeps Most of the GrowthWhy does half Kelly often look like an attractive compromise in practice?金融与交易中等essay未尝试面试订阅4088Why Fractional Kelly Is Really About Model RiskWhy is fractional Kelly best interpreted as a response to model risk rather than a rejection of Kelly logic?金融与交易中等essay未尝试面试订阅