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Graduate practice chapters

1. Graduate · Chapter 1

   Contraction estimates · Contraction slope q=-0.8; 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=-0.8; Constant b=-2.

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2. Graduate · Chapter 2

   Smoothing a weak derivative · Smoothing width ε=0.05; 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=0.05; Corner location b=-1.

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3. Graduate · Chapter 3

   Critical norm scaling · Final spatial factor λ=0.25; Lebesgue exponent p=1

   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.

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4. Graduate · Chapter 4

   Contraction estimates · Contraction slope q=-0.7; 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=-0.7; Constant b=-1.5.

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5. Graduate · Chapter 5

   Smoothing a weak derivative · Smoothing width ε=0.1; 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=0.1; Corner location b=-0.75.

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6. Graduate · Chapter 6

   Heat semigroup modes · Higher-mode amplitude a=0.4; 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=0.4; Higher frequency k=3.

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7. Graduate · Chapter 7

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

   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=0.5.

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8. Graduate · Chapter 8

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

   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=0.4.

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9. Graduate · Chapter 9

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

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10. Graduate · Chapter 10

    Critical norm scaling · Final spatial factor λ=0.5; Lebesgue exponent p=1.5

    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=1.5.

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11. Graduate · Chapter 11

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

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12. Graduate · Chapter 12

    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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13. Graduate · Chapter 13

    Contraction estimates · Contraction slope q=-0.6; Constant b=-1

    Use n=7 iterations. Find the fixed point, seventh iterate, and exact absolute error from the fixed point. Givens: Contraction slope q=-0.6; Constant b=-1.

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14. Graduate · Chapter 14

    Smoothing a weak derivative · Smoothing width ε=0.15; 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=0.15; Corner location b=-0.5.

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15. Graduate · Chapter 15

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

    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.

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16. Graduate · Chapter 16

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

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17. Graduate · Chapter 17

    Critical norm scaling · Final spatial factor λ=0.75; Lebesgue exponent p=2

    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.

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18. Graduate · Chapter 18

    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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19. Graduate · Chapter 19

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

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20. Graduate · Chapter 20

    Smoothing a weak derivative · Smoothing width ε=0.2; 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=0.2; Corner location b=-0.25.

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