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It is known that ζ(3) is irrational (Apéry's theorem) and that infinitely many of the numbers ζ(2n + 1) : n ∈ , are irrational. [1] There are also results on the irrationality of values of the Riemann zeta function at the elements of certain subsets of the positive odd integers; for example, at least one of ζ (5), ζ (7), ζ (9), or ζ ...
In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, [ a] which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size. With the restricted definition, each Farey sequence starts with the value 0, denoted ...
For a vertical line, this is 1 : 0, a kind of division by zero. In another interpretation, the quotient Q {\displaystyle Q} represents the ratio N : D . {\displaystyle N:D.} [ 6 ] For example, a cake recipe might call for ten cups of flour and two cups of sugar, a ratio of 10 : 2 {\displaystyle 10:2} or, proportionally, 5 : 1. {\displaystyle 5:1.}
Network science. In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them. [1][2] The theory of random graphs lies at the intersection between graph theory and probability theory.
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.
In mathematics, the term linear function refers to two distinct but related notions: [1] In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. [2] For distinguishing such a linear function from the other concept, the term affine function is often used.
For instance, the diamond graph K 1,1,2 (two triangles sharing an edge) has four graph automorphisms but its line graph K 1,2,2 has eight. In the illustration of the diamond graph shown, rotating the graph by 90 degrees is not a symmetry of the graph, but is a symmetry of its line graph.
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, [1] and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. [2] Since the problem had withstood the attacks of ...