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The most direct solution to a word problem takes the form of a normal form theorem and algorithm which maps every element in an equivalence class of expressions to a single encoding known as the normal form - the word problem is then solved by comparing these normal forms via syntactic equality. [1]
Then the word problem in is solvable: given two words , in the generators of , write them as words in and compare them using the solution to the word problem in . It is easy to think that this demonstrates a uniform solution of the word problem for the class K {\displaystyle K} (say) of finitely generated groups that can be embedded in G ...
A simplicial 3-complex. In mathematics, a simplicial complex is a structured set composed of points, line segments, triangles, and their n-dimensional counterparts, called simplices, such that all the faces and intersections of the elements are also included in the set (see illustration).
That is, there is an algorithm that can decide whether any given simplicial complex is homeomorphic to the boundary of a 4-dimensional ball. The recognition problem is undecidable for the d-dimensional sphere for any d ≥ 5. The proof is by reduction from the word problem for groups. From this, it can be proved that the recognition problem is ...
A simple computation shows that the map + is a ring isomorphism from the field of complex numbers to the ring of these matrices, proving that these matrices form a field. This isomorphism associates the square of the absolute value of a complex number with the determinant of the corresponding matrix, and the conjugate of a complex number with ...
The complexity class P (all problems solvable, deterministically, in polynomial time) is contained in NP (problems where solutions can be verified in polynomial time), because if a problem is solvable in polynomial time, then a solution is also verifiable in polynomial time by simply solving the problem.
Jim Simons, the legendary "Quant King" who founded Renaissance Technologies, died Friday at the age of 86, after forever changing Wall Street with his genius for math and finding patterns in data.
A word wall is a literacy tool composed of an organized collection of vocabulary words that are displayed in large visible letters on a wall, bulletin board, or other display surface in a classroom. The word wall is designed to be an interactive tool for students or others to use, and contains an array of words that can be used during writing ...