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Just as in chemistry, the characteristic property of a material will serve to identify a sample, or in the study of materials, structures and properties will determine characterization, in mathematics there is a continual effort to express properties that will distinguish a desired feature in a theory or system. Characterization is not unique ...
A three-dimensional model of a figure-eight knot.The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4 1.. Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling ...
Non-Archimedean local fields of characteristic p (for p any given prime number): the field of formal Laurent series F q ((T)) over a finite field F q, where q is a power of p. In particular, of importance in number theory, classes of local fields show up as the completions of algebraic number fields with respect to their discrete valuation ...
In mathematics, the Chern–Simons forms are certain secondary characteristic classes. [1] The theory is named for Shiing-Shen Chern and James Harris Simons , co-authors of a 1974 paper entitled "Characteristic Forms and Geometric Invariants," from which the theory arose.
In mathematics, a characteristic class is a way of associating to each principal bundle of X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" and whether it possesses sections .
Purely inseparable extensions do occur naturally; for example, they occur in algebraic geometry over fields of prime characteristic. If K is a field of characteristic p , and if V is an algebraic variety over K of dimension greater than zero, the function field K ( V ) is a purely inseparable extension over the subfield K ( V ) p of p th powers ...
In the example above, the discriminant of the number field () with x 3 − x − 1 = 0 is −23, and as we have seen the 23-adic place ramifies. The Dedekind discriminant tells us it is the only ultrametric place that does.
then F is said to have characteristic 0. [11] For example, the field of rational numbers Q has characteristic 0 since no positive integer n is zero. Otherwise, if there is a positive integer n satisfying this equation, the smallest such positive integer can be shown to be a prime number.
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