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Graduate practice chapters

1. Graduate · Chapter 221

   Heat semigroup modes · Higher-mode amplitude a=1.6; 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=1.6; Higher frequency k=6.

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2. Graduate · Chapter 222

   Variational energy · Initial sine amplitude a=-0.5; Deviation penalty b=3

   Use the remaining amplitude A=-0.2, 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=-0.5; Deviation penalty b=3.

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3. Graduate · Chapter 223

   Coercivity in a finite model · First diagonal entry a=2.6; 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=2.6; Second diagonal entry b=2.8.

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4. Graduate · Chapter 224

   Critical norm scaling · Final spatial factor λ=3.5; Lebesgue exponent p=2.5

   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=2.5.

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5. Graduate · Chapter 225

   Rare events and convergence · Magnitude exponent a=2; 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=2; Rarity exponent b=2.75.

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6. Graduate · Chapter 226

   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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7. Graduate · Chapter 227

   Contraction estimates · Contraction slope q=0.3; 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=0.3; Constant b=-0.5.

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8. Graduate · Chapter 228

   Smoothing a weak derivative · Smoothing width ε=0.45; 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=0.45; Corner location b=-1.

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9. Graduate · Chapter 229

   Heat semigroup modes · Higher-mode amplitude a=1.8; Higher frequency k=2

   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=2.

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10. Graduate · Chapter 230

    Variational energy · Initial sine amplitude a=0.5; Deviation penalty b=0.5

    Use the remaining amplitude A=0.2, 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=0.5; Deviation penalty b=0.5.

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11. Graduate · Chapter 231

    Coercivity in a finite model · First diagonal entry a=2.8; 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=2.8; Second diagonal entry b=3.

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12. Graduate · Chapter 232

    Critical norm scaling · Final spatial factor λ=3.75; Lebesgue exponent p=3

    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=3.

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13. Graduate · Chapter 233

    Rare events and convergence · Magnitude exponent a=0.5; Rarity exponent b=1.5

    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=1.5.

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14. Graduate · Chapter 234

    Conditional expectation by groups · First group baseline a=1; 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=1; Second group baseline b=2.

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15. Graduate · Chapter 235

    Contraction estimates · Contraction slope q=0.4; 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=0.4; Constant b=0.

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16. Graduate · Chapter 236

    Smoothing a weak derivative · Smoothing width ε=0.5; 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=0.5; Corner location b=-0.75.

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17. Graduate · Chapter 237

    Heat semigroup modes · Higher-mode amplitude a=2; 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=2; Higher frequency k=3.

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18. Graduate · Chapter 238

    Variational energy · Initial sine amplitude a=1; Deviation penalty b=1

    Use the remaining amplitude A=0.4, 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; Deviation penalty b=1.

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19. Graduate · Chapter 239

    Critical norm scaling · Final spatial factor λ=4; Lebesgue exponent p=3.5

    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=3.5.

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20. Graduate · Chapter 240

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

    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=1.75.

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