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Graduate practice chapters

1. Graduate · Chapter 401

   Coercivity in a finite model · First diagonal entry a=1.4; 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=1.4; Second diagonal entry b=2.

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2. Graduate · Chapter 402

   Critical norm scaling · Final spatial factor λ=2; Lebesgue exponent p=6

   Evaluate at the current scale λ=1.6. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=2; Lebesgue exponent p=6.

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3. Graduate · Chapter 403

   Rare events and convergence · Magnitude exponent a=1; 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; Rarity exponent b=2.75.

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4. Graduate · Chapter 404

   Conditional expectation by groups · First group baseline a=1.5; 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=1.5; Second group baseline b=-2.

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5. Graduate · Chapter 405

   Contraction estimates · Contraction slope q=-0.7; 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=-0.7; Constant b=0.

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6. Graduate · Chapter 406

   Smoothing a weak derivative · Smoothing width ε=0.65; 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=0.65; Corner location b=0.25.

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7. Graduate · Chapter 407

   Coercivity in a finite model · First diagonal entry a=1.6; 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=1.6; Second diagonal entry b=2.2.

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8. Graduate · Chapter 408

   Critical norm scaling · Final spatial factor λ=2.25; Lebesgue exponent p=1

   Evaluate at the current scale λ=1.75. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=2.25; Lebesgue exponent p=1.

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9. Graduate · Chapter 409

   Rare events and convergence · Magnitude exponent a=1.25; Rarity exponent b=3

   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=3.

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10. Graduate · Chapter 410

    Conditional expectation by groups · First group baseline a=2; 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=2; Second group baseline b=-1.5.

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11. Graduate · Chapter 411

    Contraction estimates · Contraction slope q=-0.6; 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=-0.6; Constant b=0.5.

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12. Graduate · Chapter 412

    Smoothing a weak derivative · Smoothing width ε=0.7; 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=0.7; Corner location b=0.5.

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13. Graduate · Chapter 413

    Variational energy · Initial sine amplitude a=1; Deviation penalty b=2.5

    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=2.5.

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14. Graduate · Chapter 414

    Coercivity in a finite model · First diagonal entry a=1.8; Second diagonal entry b=2.4

    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=2.4.

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15. Graduate · Chapter 415

    Critical norm scaling · Final spatial factor λ=2.5; Lebesgue exponent p=1.5

    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=1.5.

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16. Graduate · Chapter 416

    Conditional expectation by groups · First group baseline a=2.5; 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=2.5; Second group baseline b=-1.

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17. Graduate · Chapter 417

    Contraction estimates · Contraction slope q=-0.5; 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=-0.5; Constant b=1.

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18. Graduate · Chapter 418

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

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19. Graduate · Chapter 419

    Variational energy · Initial sine amplitude a=1.5; Deviation penalty b=3

    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=3.

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20. Graduate · Chapter 420

    Coercivity in a finite model · First diagonal entry a=2; Second diagonal entry b=2.6

    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=2.6.

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