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The inequalities then follow easily by the Pythagorean theorem. Comparison of harmonic, geometric, arithmetic, quadratic and other mean values of two positive real numbers x 1 {\displaystyle x_{1}} and x 2 {\displaystyle x_{2}}
The S-procedure or S-lemma is a mathematical result that gives conditions under which a particular quadratic inequality is a consequence of another quadratic inequality. The S-procedure was developed independently in a number of different contexts [1] [2] and has applications in control theory, linear algebra and mathematical optimization.
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.
In real algebraic geometry, the Łojasiewicz inequality, named after Stanisław Łojasiewicz, gives an upper bound for the distance of a point to the nearest zero of a given real analytic function. Specifically, let ƒ : U → R be a real analytic function on an open set U in R n , and let Z be the zero locus of ƒ.
Real algebra is the part of algebra which is relevant to real algebraic (and semialgebraic) geometry. It is mostly concerned with the study of ordered fields and ordered rings (in particular real closed fields ) and their applications to the study of positive polynomials and sums-of-squares of polynomials .
Since by the inequality of arithmetic and geometric means, this shows for the n = 2 case that H ≤ G (a property that in fact holds for all n). It also follows that G = A H {\displaystyle G={\sqrt {AH}}} , meaning the two numbers' geometric mean equals the geometric mean of their arithmetic and harmonic means.
The resulting identity is one of the most commonly used in mathematics. Among many uses, it gives a simple proof of the AM–GM inequality in two variables. The proof holds in any commutative ring. Conversely, if this identity holds in a ring R for all pairs of elements a and b, then R is commutative. To see this, apply the distributive law to ...
An important application of the Riccati equation is to the 3rd order Schwarzian differential equation ():= (″ ′) ′ (″ ′) = which occurs in the theory of conformal mapping and univalent functions. In this case the ODEs are in the complex domain and differentiation is with respect to a complex variable.