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Graduate practice chapters

1. Graduate · Chapter 181

   Coercivity in a finite model · First diagonal entry a=1.6; Second diagonal entry b=1.8

   Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=1.6; Second diagonal entry b=1.8.

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2. Graduate · Chapter 182

   Fourier incompressibility projection · Frequency component k₁=2; Frequency component k₂=1.5

   Use θ=6π/5 radians. Compute the projected coefficient Pₖv=v−k(k·v)/|k|², its dot product with k, and its norm. Givens: Frequency component k₁=2; Frequency component k₂=1.5.

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3. Graduate · Chapter 183

   Critical norm scaling · Final spatial factor λ=2.25; Lebesgue exponent p=5.5

   Evaluate at the current scale λ=1.75. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=2.25; Lebesgue exponent p=5.5.

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4. Graduate · Chapter 184

   Rare events and convergence · Magnitude exponent a=0.75; Rarity exponent b=1.5

   Use n=18 for the finite calculation, then consider n→∞. Find the nonzero probability, nonzero size, and mean absolute value; classify convergence of that mean to zero. Givens: Magnitude exponent a=0.75; Rarity exponent b=1.5.

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5. Graduate · Chapter 185

   Conditional expectation by groups · First group baseline a=-2; Second group baseline b=-1

   Use the completed group estimates at the animation endpoint. Find E[X|G] on both groups, E[X], and the mean squared prediction error. Givens: First group baseline a=-2; Second group baseline b=-1.

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6. Graduate · Chapter 186

   Contraction estimates · Contraction slope q=-0.2; Constant b=1.5

   Use n=7 iterations. Find the fixed point, seventh iterate, and exact absolute error from the fixed point. Givens: Contraction slope q=-0.2; Constant b=1.5.

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7. Graduate · Chapter 187

   Smoothing a weak derivative · Smoothing width ε=0.2; Corner location b=0

   Evaluate at x=0.6. Find the smooth derivative, the sign-function representative of the weak derivative, and the pointwise smoothing error. Givens: Smoothing width ε=0.2; Corner location b=0.

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8. Graduate · Chapter 188

   Heat semigroup modes · Higher-mode amplitude a=0.8; Higher frequency k=2

   Use time t=1.8. Find the two Fourier amplitudes and the spatial mean of u²/2. Givens: Higher-mode amplitude a=0.8; Higher frequency k=2.

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9. Graduate · Chapter 189

   Variational energy · Initial sine amplitude a=2; Deviation penalty b=1.5

   Use the remaining amplitude A=0.8, two fifths of the initial amplitude. Calculate E[u_A], its derivative with respect to A, and the minimum within this trial family. Givens: Initial sine amplitude a=2; Deviation penalty b=1.5.

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10. Graduate · Chapter 190

    Coercivity in a finite model · First diagonal entry a=1.8; Second diagonal entry b=2

    Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=1.8; Second diagonal entry b=2.

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11. Graduate · Chapter 191

    Critical norm scaling · Final spatial factor λ=2.5; Lebesgue exponent p=6

    Evaluate at the current scale λ=1.9. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=2.5; Lebesgue exponent p=6.

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12. Graduate · Chapter 192

    Rare events and convergence · Magnitude exponent a=1; Rarity exponent b=1.75

    Use n=18 for the finite calculation, then consider n→∞. Find the nonzero probability, nonzero size, and mean absolute value; classify convergence of that mean to zero. Givens: Magnitude exponent a=1; Rarity exponent b=1.75.

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13. Graduate · Chapter 193

    Beta–Bernoulli updating · Prior shape α=4; Prior shape β=3

    Use the first six fixed observations: 1, 1, 0, 1, 0, 0. Find the posterior shape parameters and the predictive probability of success on the next trial. Givens: Prior shape α=4; Prior shape β=3.

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14. Graduate · Chapter 194

    Contraction estimates · Contraction slope q=-0.1; Constant b=2

    Use n=7 iterations. Find the fixed point, seventh iterate, and exact absolute error from the fixed point. Givens: Contraction slope q=-0.1; Constant b=2.

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15. Graduate · Chapter 195

    Smoothing a weak derivative · Smoothing width ε=0.25; Corner location b=0.25

    Evaluate at x=0.6. Find the smooth derivative, the sign-function representative of the weak derivative, and the pointwise smoothing error. Givens: Smoothing width ε=0.25; Corner location b=0.25.

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16. Graduate · Chapter 196

    Variational energy · Initial sine amplitude a=-2; Deviation penalty b=1.5

    Use the remaining amplitude A=-0.8, two fifths of the initial amplitude. Calculate E[u_A], its derivative with respect to A, and the minimum within this trial family. Givens: Initial sine amplitude a=-2; Deviation penalty b=1.5.

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17. Graduate · Chapter 197

    Coercivity in a finite model · First diagonal entry a=2; Second diagonal entry b=2.2

    Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=2; Second diagonal entry b=2.2.

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18. Graduate · Chapter 198

    Fourier incompressibility projection · Frequency component k₁=3; Frequency component k₂=2.5

    Use θ=6π/5 radians. Compute the projected coefficient Pₖv=v−k(k·v)/|k|², its dot product with k, and its norm. Givens: Frequency component k₁=3; Frequency component k₂=2.5.

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19. Graduate · Chapter 199

    Critical norm scaling · Final spatial factor λ=2.75; Lebesgue exponent p=1

    Evaluate at the current scale λ=2.05. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=2.75; Lebesgue exponent p=1.

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20. Graduate · Chapter 200

    Rare events and convergence · Magnitude exponent a=1.25; Rarity exponent b=2

    Use n=18 for the finite calculation, then consider n→∞. Find the nonzero probability, nonzero size, and mean absolute value; classify convergence of that mean to zero. Givens: Magnitude exponent a=1.25; Rarity exponent b=2.

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