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Graduate practice chapters

1. Graduate · Chapter 261

   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 262

   Smoothing a weak derivative · Smoothing width ε=0.7; 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=0.7; Corner location b=0.25.

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3. Graduate · Chapter 263

   Heat semigroup modes · Higher-mode amplitude a=0.8; 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=0.8; Higher frequency k=3.

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4. Graduate · Chapter 264

   Variational energy · Initial sine amplitude a=-1.5; Deviation penalty b=2.5

   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=2.5.

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5. Graduate · Chapter 265

   Coercivity in a finite model · First diagonal entry a=0.8; 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=0.8; Second diagonal entry b=1.2.

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6. Graduate · Chapter 266

   Critical norm scaling · Final spatial factor λ=1.25; Lebesgue exponent p=4

   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=4.

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7. Graduate · Chapter 267

   Rare events and convergence · Magnitude exponent a=1.75; Rarity exponent b=2.75

   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=2.75.

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8. Graduate · Chapter 268

   Conditional expectation by groups · First group baseline a=-2; 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=-2; Second group baseline b=-0.5.

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9. Graduate · Chapter 269

   Contraction estimates · Contraction slope q=-0.8; 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=-0.8; Constant b=-1.

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10. Graduate · Chapter 270

    Smoothing a weak derivative · Smoothing width ε=0.75; 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=0.75; Corner location b=0.5.

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11. Graduate · Chapter 271

    Heat semigroup modes · Higher-mode amplitude a=1; 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; Higher frequency k=4.

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12. Graduate · Chapter 272

    Variational energy · Initial sine amplitude a=-1; Deviation penalty b=3

    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=3.

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13. Graduate · Chapter 273

    Critical norm scaling · Final spatial factor λ=1.5; Lebesgue exponent p=4.5

    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=4.5.

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14. Graduate · Chapter 274

    Conditional expectation by groups · First group baseline a=-1.5; 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=-1.5; Second group baseline b=0.

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15. Graduate · Chapter 275

    Contraction estimates · Contraction slope q=-0.7; 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=-0.7; Constant b=-0.5.

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16. Graduate · Chapter 276

    Smoothing a weak derivative · Smoothing width ε=0.8; 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=0.8; Corner location b=0.75.

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17. Graduate · Chapter 277

    Heat semigroup modes · Higher-mode amplitude a=1.2; 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=1.2; Higher frequency k=5.

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18. Graduate · Chapter 278

    Variational energy · Initial sine amplitude a=-0.5; Deviation penalty b=0

    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=0.

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19. Graduate · Chapter 279

    Coercivity in a finite model · First diagonal entry a=1.2; 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=1.2; Second diagonal entry b=1.6.

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20. Graduate · Chapter 280

    Critical norm scaling · Final spatial factor λ=1.75; Lebesgue exponent p=5

    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=5.

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