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Graduate practice chapters

1. Graduate · Chapter 421

   Critical norm scaling · Final spatial factor λ=2.75; Lebesgue exponent p=2

   Evaluate at the current scale λ=2.05. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=2.75; Lebesgue exponent p=2.

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2. Graduate · Chapter 422

   Rare events and convergence · Magnitude exponent a=1.75; 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=1.75; Rarity exponent b=0.75.

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3. Graduate · Chapter 423

   Conditional expectation by groups · First group baseline a=3; 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=3; Second group baseline b=-0.5.

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4. Graduate · Chapter 424

   Contraction estimates · Contraction slope q=-0.4; 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=-0.4; Constant b=1.5.

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5. Graduate · Chapter 425

   Smoothing a weak derivative · Smoothing width ε=0.8; 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=0.8; Corner location b=1.

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6. Graduate · Chapter 426

   Coercivity in a finite model · First diagonal entry a=2.2; Second diagonal entry b=2.8

   Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=2.2; Second diagonal entry b=2.8.

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7. Graduate · Chapter 427

   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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8. Graduate · Chapter 428

   Contraction estimates · Contraction slope q=-0.3; 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=-0.3; Constant b=2.

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9. Graduate · Chapter 429

   Smoothing a weak derivative · Smoothing width ε=0.85; 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=0.85; Corner location b=-1.

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10. Graduate · Chapter 430

    Coercivity in a finite model · First diagonal entry a=2.4; Second diagonal entry b=3

    Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=2.4; Second diagonal entry b=3.

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11. Graduate · Chapter 431

    Critical norm scaling · Final spatial factor λ=3.25; Lebesgue exponent p=3

    Evaluate at the current scale λ=2.35. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=3.25; Lebesgue exponent p=3.

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12. Graduate · Chapter 432

    Rare events and convergence · Magnitude exponent a=0.5; 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=0.5; Rarity exponent b=2.5.

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13. Graduate · Chapter 433

    Conditional expectation by groups · First group baseline a=-1.5; 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=-1.5; Second group baseline b=1.

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14. Graduate · Chapter 434

    Contraction estimates · Contraction slope q=-0.2; 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=-0.2; Constant b=-2.

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15. Graduate · Chapter 435

    Smoothing a weak derivative · Smoothing width ε=0.9; 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=0.9; Corner location b=-0.75.

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16. Graduate · Chapter 436

    Coercivity in a finite model · First diagonal entry a=2.6; 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=2.6; Second diagonal entry b=0.2.

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17. Graduate · Chapter 437

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

    Evaluate at the current scale λ=2.5. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=3.5; Lebesgue exponent p=3.5.

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18. Graduate · Chapter 438

    Rare events and convergence · Magnitude exponent a=0.75; 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=0.75; Rarity exponent b=2.75.

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19. Graduate · Chapter 439

    Conditional expectation by groups · First group baseline a=-1; 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; Second group baseline b=1.5.

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20. Graduate · Chapter 440

    Contraction estimates · Contraction slope q=-0.1; 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=-0.1; Constant b=-1.5.

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