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Graduate practice chapters

1. Graduate · Chapter 341

   Coercivity in a finite model · First diagonal entry a=2.8; Second diagonal entry b=0.2

   Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=2.8; Second diagonal entry b=0.2.

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2. Graduate · Chapter 342

   Critical norm scaling · Final spatial factor λ=3.75; Lebesgue exponent p=3.5

   Evaluate at the current scale λ=2.65. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=3.75; Lebesgue exponent p=3.5.

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3. Graduate · Chapter 343

   Rare events and convergence · Magnitude exponent a=0.75; Rarity exponent b=2.25

   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=0.75; Rarity exponent b=2.25.

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4. Graduate · Chapter 344

   Conditional expectation by groups · First group baseline a=3; 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=3; Second group baseline b=-1.

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5. Graduate · Chapter 345

   Contraction estimates · Contraction slope q=0.2; 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=0.2; Constant b=-0.5.

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6. Graduate · Chapter 346

   Smoothing a weak derivative · Smoothing width ε=0.25; 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=0.25; Corner location b=0.5.

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7. Graduate · Chapter 347

   Heat semigroup modes · Higher-mode amplitude a=1; 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; Higher frequency k=5.

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8. Graduate · Chapter 348

   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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9. Graduate · Chapter 349

   Coercivity in a finite model · First diagonal entry a=3; 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=3; Second diagonal entry b=0.4.

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10. Graduate · Chapter 350

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

    Evaluate at the current scale λ=2.8. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=4; Lebesgue exponent p=4.

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11. Graduate · Chapter 351

    Rare events and convergence · Magnitude exponent a=1; Rarity exponent b=2.5

    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; Rarity exponent b=2.5.

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12. Graduate · Chapter 352

    Conditional expectation by groups · First group baseline a=-2; 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=-2; Second group baseline b=0.

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13. Graduate · Chapter 353

    Contraction estimates · Contraction slope q=0.3; 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=0.3; Constant b=0.

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14. Graduate · Chapter 354

    Smoothing a weak derivative · Smoothing width ε=0.3; 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=0.3; Corner location b=0.75.

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15. Graduate · Chapter 355

    Heat semigroup modes · Higher-mode amplitude a=1.2; Higher frequency k=6

    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=6.

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16. Graduate · Chapter 356

    Variational energy · Initial sine amplitude a=1; Deviation penalty b=2

    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=2.

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17. Graduate · Chapter 357

    Coercivity in a finite model · First diagonal entry a=0.2; 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=0.2; Second diagonal entry b=0.8.

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18. Graduate · Chapter 358

    Critical norm scaling · Final spatial factor λ=0.25; Lebesgue exponent p=2.5

    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=2.5.

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19. Graduate · Chapter 359

    Rare events and convergence · Magnitude exponent a=1.25; 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=1.25; Rarity exponent b=2.75.

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20. Graduate · Chapter 360

    Conditional expectation by groups · First group baseline a=-1.5; 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=-1.5; Second group baseline b=0.5.

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