Search results
Results From The WOW.Com Content Network
Another special case is the class of problems where each clause contains XOR (i.e. exclusive or) rather than (plain) OR operators. [e] This is in P, since an XOR-SAT formula can also be viewed as a system of linear equations mod 2, and can be solved in cubic time by Gaussian elimination; [18] see the box for an example.
[4]: 23–24 The pair held practice sessions, in which the problems were put to university students and worked through as a class (with some of the representative problems solved by the teacher, and the harder problems set as homework). They went through portions of the book at a rate of about one chapter a semester. [7]: xi–xii
The PCP theorem states that NP = PCP[O(log n), O(1)],. where PCP[r(n), q(n)] is the class of problems for which a probabilistically checkable proof of a solution can be given, such that the proof can be checked in polynomial time using r(n) bits of randomness and by reading q(n) bits of the proof, correct proofs are always accepted, and incorrect proofs are rejected with probability at least 1/2.
In 1983, Trahtman solved this problem by proving that all semigroups of order less than six are finitely based. [15] [16] In the theory of varieties of semigroups and universal algebras the problem of existence of covering elements in the lattice of varieties was posed by Evans in 1971. [17] The positive solution of the problem was found by ...
Problems 1, 2, 5, 6, [g] 9, 11, 12, 15, 21, and 22 have solutions that have partial acceptance, but there exists some controversy as to whether they resolve the problems. That leaves 8 (the Riemann hypothesis), 13 and 16 [h] unresolved, and 4 and 23 as too vague to ever be described as solved. The withdrawn 24 would also be in this class.
An answer to the P versus NP question would determine whether problems that can be verified in polynomial time can also be solved in polynomial time. If P ≠ NP, which is widely believed, it would mean that there are problems in NP that are harder to compute than to verify: they could not be solved in polynomial time, but the answer could be ...
In 1922, Cole published his paper Kirkman Parades [4] which listed for the first time all seven non-isomorphic solutions to the 15 schoolgirl problem, thus answering a long-standing question since the 1850s. The seven Kirkman solutions correspond to four different Steiner systems when resolvability into parallel classes is removed as a constraint.
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.