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A word equation is a formal equality:= = between a pair of words and , each over an alphabet comprising both constants (c.f. ) and unknowns (c.f. ). [1] An assignment of constant words to the unknowns of is said to solve if it maps both sides of to identical words.
In layman's terms, the genus is the number of "holes" an object has ("holes" interpreted in the sense of doughnut holes; a hollow sphere would be considered as having zero holes in this sense). [3] A torus has 1 such hole, while a sphere has 0. The green surface pictured above has 2 holes of the relevant sort. For instance:
With finite automata, the edges are labeled with a letter in an alphabet. To use the graph, one starts at a node and travels along the edges to reach a final node. The path taken along the graph forms the word. It is a finite graph because there are a countable number of nodes and edges, and only one path connects two distinct nodes. [1]
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.
Wheel graphs W 2n+1, for n ≥ 2, are not word-representable and W 5 is the minimum (by the number of vertices) non-word-representable graph. Taking any non-comparability graph and adding an apex (a vertex connected to any other vertex), we obtain a non-word-representable graph, which then can produce infinitely many non-word-representable ...
For a torus, the first Betti number is b 1 = 2 , which can be intuitively thought of as the number of circular "holes" Informally, the kth Betti number refers to the number of k-dimensional holes on a topological surface. A "k-dimensional hole" is a k-dimensional cycle that is not a boundary of a (k+1)-dimensional object.
This is a graph with vertex set the pants decompositions of , and two vertices are joined if they are related by an elementary move, which is one of the two following operations: take a curve α {\displaystyle \alpha } in the decomposition in a one-holed torus and replace it by a curve in the torus intersecting it only once,
The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics. Additionally, the subsequent columns contains an informal explanation, a short example, the Unicode location, the name for use in HTML documents, [1] and the LaTeX symbol.