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Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
Many computer systems measure time and date using Unix time, an international standard for digital timekeeping. Unix time is defined as the number of seconds elapsed since 00:00:00 UTC on 1 January 1970 (an arbitrarily chosen time based on the creation of the first Unix system), which has been dubbed the Unix epoch. [6]
The other six Millennium Prize Problems remain unsolved, despite a large number of unsatisfactory proofs by both amateur and professional mathematicians. Andrew Wiles , as part of the Clay Institute's scientific advisory board, hoped that the choice of US$ 1 million prize money would popularize, among general audiences, both the selected ...
Determining whether a Turing machine is a busy beaver champion (i.e., is the longest-running among halting Turing machines with the same number of states and symbols). Rice's theorem states that for all nontrivial properties of partial functions, it is undecidable whether a given machine computes a partial function with that property.
Can graphs of bounded clique-width be recognized in polynomial time? [3] Can one find a simple closed quasigeodesic on a convex polyhedron in polynomial time? [4] Can a simultaneous embedding with fixed edges for two given graphs be found in polynomial time? [5] Can the square-root sum problem be solved in polynomial time in the Turing machine ...
Prisoner 3 opens drawers 3 and 6, where they find their own number. Prisoner 4 opens drawers 4, 8, and 2, where they find their own number. This is the same cycle that was encountered by prisoner 2 and will be encountered by prisoner 8. Each of these prisoners will find their own number in the third opened drawer.
A decision problem is EXPTIME-complete if it is in EXPTIME, and every problem in EXPTIME has a polynomial-time many-one reduction to it. A number of problems are known to be EXPTIME-complete. Because it can be shown that P ≠ EXPTIME, these problems are outside P, and so require more than polynomial time.
To achieve an O(n 2) running time, a ranking matrix whose entry at row i and column j is the position of the jth individual in the ith's list; this takes O(n 2) time. With the ranking matrix, checking whether an individual prefers one to another can be done in constant time by comparing their ranks in the matrix.