nenegativita

Nenegativita, in a mathematical context, refers to nonnegativity—the property of a quantity being greater than or equal to zero. It is a foundational idea across several disciplines, including algebra, analysis, optimization, probability, and statistics, where quantities are often required to have nonnegative values for meaningful interpretation or feasibility.

A precise formulation can refer to different objects. For a real-valued function f defined on a domain X, nonnegativity means f(x) ≥ 0 for all x in X. For a vector x in R^n, nonnegativity means each component x_i ≥ 0. For a matrix A, nonnegativity typically means every entry a_ij ≥ 0. Related notions distinguish nonnegativity from positivity, where strictly positive means all components or entries are greater than zero.

Common examples illustrate the concept. The function f(x) = x^2 is nonnegative for all real x. A vector (2, 0, 5) is nonnegative, as all components are nonnegative. A nonnegative matrix, such as one with all entries in the interval [0, ∞), arises in various models, including Markov chains and certain economic or network problems.

Properties and implications follow from the definition. Nonnegativity is preserved under addition and under multiplication by nonnegative scalars. However, subtraction of nonnegative quantities is not guaranteed to be nonnegative. In optimization, nonnegativity constraints define feasible regions, while in probability, distributions are modeled as nonnegative and often normalized to sum to one.

See also: nonnegativity constraint, nonnegative matrix, probability distribution, positivity (mathematics).

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