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Graduate practice chapters

1. Graduate · Chapter 121

   Critical norm scaling · Final spatial factor λ=0.25; Lebesgue exponent p=1.5

   Evaluate at the current scale λ=0.55. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=0.25; Lebesgue exponent p=1.5.

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2. Graduate · Chapter 122

   Rare events and convergence · Magnitude exponent a=0.75; Rarity exponent b=1.25

   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=1.25.

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3. Graduate · Chapter 123

   Conditional expectation by groups · First group baseline a=0; Second group baseline b=0.5

   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=0; Second group baseline b=0.5.

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4. Graduate · Chapter 124

   Beta–Bernoulli updating · Prior shape α=1; Prior shape β=4

   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=1; Prior shape β=4.

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5. Graduate · Chapter 125

   Contraction estimates · Contraction slope q=0.8; 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=0.8; Constant b=1.5.

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6. Graduate · Chapter 126

   Smoothing a weak derivative · Smoothing width ε=0.85; Corner location b=0.75

   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=0.85; Corner location b=0.75.

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7. Graduate · Chapter 127

   Variational energy · Initial sine amplitude a=-2; Deviation penalty b=1

   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.

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8. Graduate · Chapter 128

   Coercivity in a finite model · First diagonal entry a=0.4; Second diagonal entry b=0.6

   Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=0.4; Second diagonal entry b=0.6.

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9. Graduate · Chapter 129

   Fourier incompressibility projection · Frequency component k₁=1.5; Frequency component k₂=3

   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=1.5; Frequency component k₂=3.

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10. Graduate · Chapter 130

    Critical norm scaling · Final spatial factor λ=0.5; Lebesgue exponent p=2

    Evaluate at the current scale λ=0.7. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=0.5; Lebesgue exponent p=2.

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11. Graduate · Chapter 131

    Rare events and convergence · Magnitude exponent a=1; 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=1; Rarity exponent b=1.5.

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12. Graduate · Chapter 132

    Conditional expectation by groups · First group baseline a=0.5; 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=0.5; Second group baseline b=1.

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13. Graduate · Chapter 133

    Contraction estimates · Contraction slope q=-0.8; 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=-0.8; Constant b=-1.5.

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14. Graduate · Chapter 134

    Smoothing a weak derivative · Smoothing width ε=0.9; Corner location b=1

    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=0.9; Corner location b=1.

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15. Graduate · Chapter 135

    Heat semigroup modes · Higher-mode amplitude a=1.6; Higher frequency k=5

    Use time t=1.8. Find the two Fourier amplitudes and the spatial mean of u²/2. Givens: Higher-mode amplitude a=1.6; Higher frequency k=5.

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16. Graduate · Chapter 136

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

    Use the remaining amplitude A=-0.6, 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=-1.5; Deviation penalty b=1.5.

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17. Graduate · Chapter 137

    Coercivity in a finite model · First diagonal entry a=0.6; Second diagonal entry b=0.8

    Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=0.6; Second diagonal entry b=0.8.

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18. Graduate · Chapter 138

    Fourier incompressibility projection · Frequency component k₁=2; Frequency component k₂=1

    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.

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19. Graduate · Chapter 139

    Critical norm scaling · Final spatial factor λ=0.75; Lebesgue exponent p=2.5

    Evaluate at the current scale λ=0.85. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=0.75; Lebesgue exponent p=2.5.

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20. Graduate · Chapter 140

    Rare events and convergence · Magnitude exponent a=1.25; 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=1.25; Rarity exponent b=1.75.

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