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Graduate practice chapters

1. Graduate · Chapter 601

   Contraction estimates · Contraction slope q=-0.3; 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=-0.3; Constant b=-1.5.

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2. Graduate · Chapter 602

   Smoothing a weak derivative · Smoothing width ε=0.5; Corner location b=0

   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=0.5; Corner location b=0.

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3. Graduate · Chapter 603

   Coercivity in a finite model · First diagonal entry a=3; 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=3; Second diagonal entry b=1.

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4. Graduate · Chapter 604

   Critical norm scaling · Final spatial factor λ=4; Lebesgue exponent p=5.5

   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=5.5.

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5. Graduate · Chapter 605

   Conditional expectation by groups · First group baseline a=-1.5; 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=-1.5; Second group baseline b=2.5.

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6. Graduate · Chapter 606

   Contraction estimates · Contraction slope q=-0.2; 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=-0.2; Constant b=-1.

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7. Graduate · Chapter 607

   Smoothing a weak derivative · Smoothing width ε=0.55; 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=0.55; Corner location b=0.25.

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8. Graduate · Chapter 608

   Coercivity in a finite model · First diagonal entry a=0.2; Second diagonal entry b=1.4

   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=1.4.

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9. Graduate · Chapter 609

   Critical norm scaling · Final spatial factor λ=0.25; Lebesgue exponent p=4

   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=4.

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10. Graduate · Chapter 610

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

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11. Graduate · Chapter 611

    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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12. Graduate · Chapter 612

    Smoothing a weak derivative · Smoothing width ε=0.6; 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=0.6; Corner location b=0.5.

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13. Graduate · Chapter 613

    Coercivity in a finite model · First diagonal entry a=0.4; 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=0.4; Second diagonal entry b=1.6.

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14. Graduate · Chapter 614

    Critical norm scaling · Final spatial factor λ=0.5; Lebesgue exponent p=4.5

    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=4.5.

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15. Graduate · Chapter 615

    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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16. Graduate · Chapter 616

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

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17. Graduate · Chapter 617

    Smoothing a weak derivative · Smoothing width ε=0.65; 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=0.65; Corner location b=0.75.

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18. Graduate · Chapter 618

    Coercivity in a finite model · First diagonal entry a=0.6; Second diagonal entry b=1.8

    Use θ=6π/5 radians. Find vᵀAv and the sharp lower and upper bounds over all unit directions. Givens: First diagonal entry a=0.6; Second diagonal entry b=1.8.

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19. Graduate · Chapter 619

    Critical norm scaling · Final spatial factor λ=0.75; Lebesgue exponent p=5

    Evaluate at the current scale λ=0.85. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=0.75; Lebesgue exponent p=5.

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20. Graduate · Chapter 620

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

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