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In mathematics, the QM-AM-GM-HM inequalities, also known as the mean inequality chain, state the relationship between the harmonic mean, geometric mean, arithmetic mean, and quadratic mean (also known as root mean square). Suppose that ,, …, are positive real numbers. Then
The word problem for an algebra is then to determine, given two expressions (words) involving the generators and operations, whether they represent the same element of the algebra modulo the identities. The word problems for groups and semigroups can be phrased as word problems for algebras. [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.
The feasible regions of linear programming are defined by a set of inequalities. In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. [1] It is used most often to compare two numbers on the number line by their size.
If the objective function is concave (maximization problem), or convex (minimization problem) and the constraint set is convex, then the program is called convex and general methods from convex optimization can be used in most cases. If the objective function is quadratic and the constraints are linear, quadratic programming techniques are used.
In general, whenever we multiply both sides of an equation by an expression involving variables, we introduce extraneous solutions wherever that expression is equal to zero. But it is not sufficient to exclude these values, because they may have been legitimate solutions to the original equation.
Bennett's inequality, an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount Bhatia–Davis inequality , an upper bound on the variance of any bounded probability distribution
Geometric programming — problems involving signomials or posynomials Signomial — similar to polynomials, but exponents need not be integers; Posynomial — a signomial with positive coefficients; Quadratically constrained quadratic program; Linear-fractional programming — objective is ratio of linear functions, constraints are linear
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