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Graduate practice chapters

1. Graduate · Chapter 561

   Coercivity in a finite model · First diagonal entry a=1.2; 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=1.2; Second diagonal entry b=2.2.

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2. Graduate · Chapter 562

   Critical norm scaling · Final spatial factor λ=1.75; Lebesgue exponent p=1

   Evaluate at the current scale λ=1.45. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=1.75; Lebesgue exponent p=1.

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3. Graduate · Chapter 563

   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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4. Graduate · Chapter 564

   Contraction estimates · Contraction slope q=0.6; 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=0.6; Constant b=-2.

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5. Graduate · Chapter 565

   Smoothing a weak derivative · Smoothing width ε=0.1; 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=0.1; Corner location b=0.25.

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6. Graduate · Chapter 566

   Coercivity in a finite model · First diagonal entry a=1.4; 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=1.4; Second diagonal entry b=2.4.

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7. Graduate · Chapter 567

   Critical norm scaling · Final spatial factor λ=2; Lebesgue exponent p=1.5

   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=1.5.

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8. Graduate · Chapter 568

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

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9. Graduate · Chapter 569

   Contraction estimates · Contraction slope q=0.7; 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=0.7; Constant b=-1.5.

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10. Graduate · Chapter 570

    Smoothing a weak derivative · Smoothing width ε=0.15; 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=0.15; Corner location b=0.5.

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11. Graduate · Chapter 571

    Coercivity in a finite model · First diagonal entry a=1.6; 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=1.6; Second diagonal entry b=2.6.

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12. Graduate · Chapter 572

    Critical norm scaling · Final spatial factor λ=2.25; Lebesgue exponent p=2

    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=2.

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13. Graduate · Chapter 573

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

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14. Graduate · Chapter 574

    Contraction estimates · Contraction slope q=0.8; 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=0.8; Constant b=-1.

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15. Graduate · Chapter 575

    Smoothing a weak derivative · Smoothing width ε=0.2; 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=0.2; Corner location b=0.75.

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16. Graduate · Chapter 576

    Coercivity in a finite model · First diagonal entry a=1.8; 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=1.8; Second diagonal entry b=2.8.

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17. Graduate · Chapter 577

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

    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=2.5.

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18. Graduate · Chapter 578

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

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19. Graduate · Chapter 579

    Contraction estimates · Contraction slope q=-0.8; 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=-0.8; Constant b=0.5.

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20. Graduate · Chapter 580

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

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