Nonnegative Polynomial
In mathematics, a positive polynomial (respectively non-negative polynomial) on a particular set (mathematics), set is a polynomial whose values are positive (respectively non-negative) on that set. Precisely, Let p be a polynomial in n variables with real number, real coefficients and let S be a subset of the n-dimensional Euclidean space \mathbb^n. We say that: * p is positive on S if p(x)>0 for every x in S. * p is non-negative on S if p(x)\ge 0 for every x in S. Positivstellensatz (and nichtnegativstellensatz) For certain sets S, there exist algebraic descriptions of all polynomials that are positive (resp. non-negative) on S. Such a description is a positivstellensatz (resp. nichtnegativstellensatz). The importance of Positivstellensatz theorems in computation arises from its ability to transform problems of polynomial optimization into semidefinite programming problems, which can be efficiently solved using convex optimization techniques. Examples of positivstellensatz ( ...
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Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ...
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