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Graduate practice chapters

1. Graduate · Chapter 441

   Smoothing a weak derivative · Smoothing width ε=0.95; 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=0.95; Corner location b=-0.5.

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2. Graduate · Chapter 442

   Coercivity in a finite model · First diagonal entry a=2.8; 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=2.8; Second diagonal entry b=0.4.

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3. Graduate · Chapter 443

   Critical norm scaling · Final spatial factor λ=3.75; Lebesgue exponent p=4

   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=4.

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4. Graduate · Chapter 444

   Rare events and convergence · Magnitude exponent a=1; 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; Rarity exponent b=3.

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5. Graduate · Chapter 445

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

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6. Graduate · Chapter 446

   Contraction estimates · Contraction slope q=0; 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; Constant b=-1.

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7. Graduate · Chapter 447

   Smoothing a weak derivative · Smoothing width ε=1; 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=1; Corner location b=-0.25.

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8. Graduate · Chapter 448

   Coercivity in a finite model · First diagonal entry a=3; 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=3; Second diagonal entry b=0.6.

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9. Graduate · Chapter 449

   Critical norm scaling · Final spatial factor λ=4; Lebesgue exponent p=4.5

   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=4.5.

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10. Graduate · Chapter 450

    Rare events and convergence · Magnitude exponent a=1.25; Rarity exponent b=0.5

    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=1.25; Rarity exponent b=0.5.

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11. Graduate · Chapter 451

    Conditional expectation by groups · First group baseline a=0; Second group baseline b=2.5

    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=0; Second group baseline b=2.5.

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12. Graduate · Chapter 452

    Contraction estimates · Contraction slope q=0.1; 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=0.1; Constant b=-0.5.

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13. Graduate · Chapter 453

    Smoothing a weak derivative · Smoothing width ε=0.05; 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=0.05; Corner location b=-0.25.

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14. Graduate · Chapter 454

    Coercivity in a finite model · First diagonal entry a=0.2; Second diagonal entry b=1

    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=1.

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15. Graduate · Chapter 455

    Rare events and convergence · Magnitude exponent a=1.5; 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=1.5; Rarity exponent b=0.75.

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16. Graduate · Chapter 456

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

    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=0.5; Second group baseline b=3.

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17. Graduate · Chapter 457

    Contraction estimates · Contraction slope q=0.2; 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=0.2; Constant b=0.

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18. Graduate · Chapter 458

    Coercivity in a finite model · First diagonal entry a=0.4; 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=0.4; Second diagonal entry b=1.2.

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19. Graduate · Chapter 459

    Critical norm scaling · Final spatial factor λ=0.5; Lebesgue exponent p=3.5

    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=3.5.

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20. Graduate · Chapter 460

    Rare events and convergence · Magnitude exponent a=1.75; Rarity exponent b=1

    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=1.

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