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Graduate practice chapters

1. Graduate · Chapter 301

   Heat semigroup modes · Higher-mode amplitude a=1.8; Higher frequency k=3

   Use time t=1.8. Find the two Fourier amplitudes and the spatial mean of u²/2. Givens: Higher-mode amplitude a=1.8; Higher frequency k=3.

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2. Graduate · Chapter 302

   Variational energy · Initial sine amplitude a=1.5; Deviation penalty b=2

   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=2.

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3. Graduate · Chapter 303

   Coercivity in a finite model · First diagonal entry a=1.8; 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=1.8; Second diagonal entry b=2.2.

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4. Graduate · Chapter 304

   Critical norm scaling · Final spatial factor λ=2.5; Lebesgue exponent p=1

   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.

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5. Graduate · Chapter 305

   Rare events and convergence · Magnitude exponent a=1.25; 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=1.25; Rarity exponent b=2.5.

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6. Graduate · Chapter 306

   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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7. Graduate · Chapter 307

   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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8. Graduate · Chapter 308

   Smoothing a weak derivative · Smoothing width ε=1; 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=1; Corner location b=-0.5.

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9. Graduate · Chapter 309

   Heat semigroup modes · Higher-mode amplitude a=2; Higher frequency k=4

   Use time t=1.8. Find the two Fourier amplitudes and the spatial mean of u²/2. Givens: Higher-mode amplitude a=2; Higher frequency k=4.

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10. Graduate · Chapter 310

    Variational energy · Initial sine amplitude a=2; Deviation penalty b=2.5

    Use the remaining amplitude A=0.8, 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=2; Deviation penalty b=2.5.

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11. Graduate · Chapter 311

    Coercivity in a finite model · First diagonal entry a=2; 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=2; Second diagonal entry b=2.4.

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12. Graduate · Chapter 312

    Critical norm scaling · Final spatial factor λ=2.75; Lebesgue exponent p=1.5

    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=1.5.

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13. Graduate · Chapter 313

    Rare events and convergence · Magnitude exponent a=1.5; 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=1.5; Rarity exponent b=2.75.

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14. Graduate · Chapter 314

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

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15. Graduate · Chapter 315

    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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16. Graduate · Chapter 316

    Smoothing a weak derivative · Smoothing width ε=0.05; 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=0.05; Corner location b=-0.5.

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17. Graduate · Chapter 317

    Heat semigroup modes · Higher-mode amplitude a=0.2; Higher frequency k=6

    Use time t=1.8. Find the two Fourier amplitudes and the spatial mean of u²/2. Givens: Higher-mode amplitude a=0.2; Higher frequency k=6.

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18. Graduate · Chapter 318

    Variational energy · Initial sine amplitude a=-2; Deviation penalty b=2.5

    Use the remaining amplitude A=-0.8, 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=-2; Deviation penalty b=2.5.

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19. Graduate · Chapter 319

    Coercivity in a finite model · First diagonal entry a=2.2; 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=2.2; Second diagonal entry b=2.6.

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20. Graduate · Chapter 320

    Critical norm scaling · Final spatial factor λ=3; Lebesgue exponent p=2

    Evaluate at the current scale λ=2.2. Find the norm-scaling exponent, the Lᵖ norm ratio, and the spatial volume ratio. Givens: Final spatial factor λ=3; Lebesgue exponent p=2.

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