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Graduate practice chapters

1. Graduate · Chapter 61

   Smoothing a weak derivative · Smoothing width ε=0.45; 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=0.45; Corner location b=1.

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2. Graduate · Chapter 62

   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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3. Graduate · Chapter 63

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

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4. Graduate · Chapter 64

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

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5. Graduate · Chapter 65

   Critical norm scaling · Final spatial factor λ=2.5; Lebesgue exponent p=5.5

   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=5.5.

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6. Graduate · Chapter 66

   Rare events and convergence · Magnitude exponent a=0.75; Rarity exponent b=1

   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.

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7. Graduate · Chapter 67

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

   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=2.

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8. Graduate · Chapter 68

   Beta–Bernoulli updating · Prior shape α=4; Prior shape β=5

   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=5.

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9. Graduate · Chapter 69

   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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10. Graduate · Chapter 70

    Smoothing a weak derivative · Smoothing width ε=0.5; 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=0.5; Corner location b=-1.

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11. Graduate · Chapter 71

    Variational energy · Initial sine amplitude a=-1.5; Deviation penalty b=1

    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.

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12. Graduate · Chapter 72

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

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13. Graduate · Chapter 73

    Critical norm scaling · Final spatial factor λ=2.75; Lebesgue exponent p=6

    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=6.

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14. Graduate · Chapter 74

    Rare events and convergence · Magnitude exponent a=1; 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=1; Rarity exponent b=1.25.

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15. Graduate · Chapter 75

    Conditional expectation by groups · First group baseline a=2.5; Second group baseline b=2.5

    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=2.5; Second group baseline b=2.5.

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16. Graduate · Chapter 76

    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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17. Graduate · Chapter 77

    Smoothing a weak derivative · Smoothing width ε=0.55; 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=0.55; Corner location b=-0.75.

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18. Graduate · Chapter 78

    Heat semigroup modes · Higher-mode amplitude a=0.2; 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=0.2; Higher frequency k=3.

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19. Graduate · Chapter 79

    Variational energy · Initial sine amplitude a=-1; Deviation penalty b=1.5

    Use the remaining amplitude A=-0.4, 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; Deviation penalty b=1.5.

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20. Graduate · Chapter 80

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

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