Problem 1
Use the precise definitions, state all hypotheses, apply theorem↔proof↔source↔open problem, and verify the conclusion against a boundary case or independent theorem.
Faculty solution guide
Expository Paper, Proof Defense and Graduate Bridge
Accept equivalent mathematically correct reasoning. Require explicit definitions, theorem hypotheses, proof structure, verification, and limitations.
Problem 2
Identify the first invalid inference, missing hypothesis, quantifier error, unjustified limit interchange, or unproved existence claim; then supply a valid repair.
Problem 3
A complete response explicitly states the definitions, domain, hypotheses, exact target, and theorem dependencies for Expository Paper, Proof Defense and Graduate Bridge.
Problem 4
The argument must address presenting a rigorous theorem sequence, limitations, and next research steps with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in geometric function theory, PDE, number theory, mathematical physics, and graduate research, verifies conditions, and names at least one limitation.
Problem 6
Accept a second valid proof, a counterexample showing necessity of a hypothesis, an independent estimate, or a reproducible finite check that is correctly described as supporting evidence.