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Graduate practice chapters

1. Graduate · Chapter 41

   Critical norm scaling · Final spatial factor λ=1.75; Lebesgue exponent p=4

   Evaluate at the current scale λ=1.45. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=1.75; Lebesgue exponent p=4.

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2. Graduate · Chapter 42

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

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3. Graduate · Chapter 43

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

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4. Graduate · Chapter 44

   Beta–Bernoulli updating · Prior shape α=1; Prior shape β=2

   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=2.

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5. Graduate · Chapter 45

   Smoothing a weak derivative · Smoothing width ε=0.35; Corner location b=0.5

   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=0.35; Corner location b=0.5.

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6. Graduate · Chapter 46

   Heat semigroup modes · Higher-mode amplitude a=1.4; Higher frequency k=3

   Use time t=1.8. Find the two Fourier amplitudes and the spatial mean of u²/2. Givens: Higher-mode amplitude a=1.4; Higher frequency k=3.

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7. Graduate · Chapter 47

   Variational energy · Initial sine amplitude a=1.5; Deviation penalty b=0

   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=0.

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8. Graduate · Chapter 48

   Coercivity in a finite model · First diagonal entry a=1.4; Second diagonal entry b=1.4

   Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=1.4; Second diagonal entry b=1.4.

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9. Graduate · Chapter 49

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

   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=2.

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10. Graduate · Chapter 50

    Critical norm scaling · Final spatial factor λ=2; Lebesgue exponent p=4.5

    Evaluate at the current scale λ=1.6. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=2; Lebesgue exponent p=4.5.

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11. Graduate · Chapter 51

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

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12. Graduate · Chapter 52

    Contraction estimates · Contraction slope q=-0.1; 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=-0.1; Constant b=1.5.

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13. Graduate · Chapter 53

    Smoothing a weak derivative · Smoothing width ε=0.4; 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=0.4; Corner location b=0.75.

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14. Graduate · Chapter 54

    Heat semigroup modes · Higher-mode amplitude a=1.6; Higher frequency k=4

    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=4.

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15. Graduate · Chapter 55

    Variational energy · Initial sine amplitude a=2; Deviation penalty b=0.5

    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=0.5.

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16. Graduate · Chapter 56

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

    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=1.6.

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17. Graduate · Chapter 57

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

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18. Graduate · Chapter 58

    Rare events and convergence · Magnitude exponent a=0.5; Rarity exponent b=0.75

    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=0.5; Rarity exponent b=0.75.

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19. Graduate · Chapter 59

    Conditional expectation by groups · First group baseline a=1.5; Second group baseline b=1.5

    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=1.5; Second group baseline b=1.5.

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20. Graduate · Chapter 60

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

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