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William Lowell Putnam Mathematical Competition results; William Lowell Putnam Competition problems, solutions, and results archive; Archive of Problems 1938–2003; Searchable data base for information about careers of Putnam Fellows; A comprehensive history of the Putnam competition An electronic update of Gallian's 2004 paper (PDF)
Theorem (Calvin Richard Putnam) [1] Let T, M, N be linear operators on a complex Hilbert space, and suppose that M and N are normal, T is bounded and MT = TN. Then M * T = TN *. First proof (Marvin Rosenblum) : By induction, the hypothesis implies that M k T = TN k for all k .
For example, Gödel–Dummett logic has a simple semantic characterization in terms of total orders. Specific intermediate logics may be given by semantical description. Others are often given by adding one or more axioms to Intuitionistic logic (usually denoted as intuitionistic propositional calculus IPC, but also Int, IL or H) Examples include:
Goodman poses Hume's problem of induction as a problem of the validity of the predictions we make. Since predictions are about what has yet to be observed and because there is no necessary connection between what has been observed and what will be observed, there is no objective justification for these predictions.
Algorithm DP SAT solver Input: A set of clauses Φ. Output: A Truth Value: true if Φ can be satisfied, false otherwise. function DP-SAT(Φ) repeat // unit propagation: while Φ contains a unit clause {l} do for every clause c in Φ that contains l do Φ ← remove-from-formula(c, Φ); for every clause c in Φ that contains ¬l do Φ ← remove-from-formula(c, Φ); Φ ← add-to-formula(c ...
As with the Hilbert problems, one of the prize problems (the Poincaré conjecture) was solved relatively soon after the problems were announced. The Riemann hypothesis is noteworthy for its appearance on the list of Hilbert problems, Smale's list, the list of Millennium Prize Problems, and even the Weil conjectures, in its geometric guise.
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