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In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. [1]
In social science, economic spectrum is used to indicate the range of social class along some indicator of wealth or income. In political science , the term political spectrum refers to a system of classifying political positions in one or more dimensions, for example in a range including right wing and left wing.
The space of bounded linear operators B(X) on a Banach space X is an example of a unital Banach algebra. Since the definition of the spectrum does not mention any properties of B(X) except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim.
In mathematics, specifically in spectral theory, a discrete spectrum of a closed linear operator is defined as the set of isolated points of its spectrum such that the rank of the corresponding Riesz projector is finite.
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory.Every such cohomology theory is representable, as follows from Brown's representability theorem.
Spectral methods and finite-element methods are closely related and built on the same ideas; the main difference between them is that spectral methods use basis functions that are generally nonzero over the whole domain, while finite element methods use basis functions that are nonzero only on small subdomains (compact support).
Spectral analysis or spectrum analysis is analysis in terms of a spectrum of frequencies or related quantities such as energies, eigenvalues, etc. In specific areas it may refer to: Spectroscopy in chemistry and physics, a method of analyzing the properties of matter from their electromagnetic interactions
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by Jean Leray (1946a, 1946b), they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra.