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Graduate practice chapters

1. Graduate · Chapter 21

   Heat semigroup modes · Higher-mode amplitude a=0.8; 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=0.8; Higher frequency k=5.

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2. Graduate · Chapter 22

   Variational energy · Initial sine amplitude a=-0.5; Deviation penalty b=1.5

   Use the remaining amplitude A=-0.2, 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=-0.5; Deviation penalty b=1.5.

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3. Graduate · Chapter 23

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

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4. Graduate · Chapter 24

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

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5. Graduate · Chapter 25

   Critical norm scaling · Final spatial factor λ=1.25; Lebesgue exponent p=3

   Evaluate at the current scale λ=1.15. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=1.25; Lebesgue exponent p=3.

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6. Graduate · Chapter 26

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

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7. Graduate · Chapter 27

   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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8. Graduate · Chapter 28

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

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9. Graduate · Chapter 29

   Contraction estimates · Contraction slope q=-0.4; Constant b=0

   Use n=7 iterations. Find the fixed point, seventh iterate, and exact absolute error from the fixed point. Givens: Contraction slope q=-0.4; Constant b=0.

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10. Graduate · Chapter 30

    Smoothing a weak derivative · Smoothing width ε=0.25; 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=0.25; Corner location b=0.

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11. Graduate · Chapter 31

    Variational energy · Initial sine amplitude a=0.5; Deviation penalty b=2.5

    Use the remaining amplitude A=0.2, 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=0.5; Deviation penalty b=2.5.

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12. Graduate · Chapter 32

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

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13. Graduate · Chapter 33

    Critical norm scaling · Final spatial factor λ=1.5; Lebesgue exponent p=3.5

    Evaluate at the current scale λ=1.3. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=1.5; Lebesgue exponent p=3.5.

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14. Graduate · Chapter 34

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

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15. Graduate · Chapter 35

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

    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.

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16. Graduate · Chapter 36

    Contraction estimates · Contraction slope q=-0.3; Constant b=0.5

    Use n=7 iterations. Find the fixed point, seventh iterate, and exact absolute error from the fixed point. Givens: Contraction slope q=-0.3; Constant b=0.5.

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17. Graduate · Chapter 37

    Smoothing a weak derivative · Smoothing width ε=0.3; 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=0.3; Corner location b=0.25.

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18. Graduate · Chapter 38

    Heat semigroup modes · Higher-mode amplitude a=1.2; 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=1.2; Higher frequency k=2.

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19. Graduate · Chapter 39

    Coercivity in a finite model · First diagonal entry a=1.2; Second diagonal entry b=1.2

    Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=1.2; Second diagonal entry b=1.2.

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20. Graduate · Chapter 40

    Fourier incompressibility projection · Frequency component k₁=1; 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=1; Frequency component k₂=1.5.

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