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In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers. [1] Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice.
A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the "imaginary unit", that satisfies i 2 = −1.
The terminology here is from complex multiplication theory, which was developed for elliptic curves in the nineteenth century. One of the major achievements in algebraic number theory and algebraic geometry of the twentieth century was to find the correct formulations of the corresponding theory for abelian varieties of dimension d > 1.
These operators are direct extensions of their complex analogs: if a and b are taken from the real subset of complex numbers, the appearance of the conjugate in the formulas has no effect, so the operators are the same as those for the complex numbers. The product of a nonzero element with its conjugate is a non-negative real number:
In mathematics, the complexification of a vector space V over the field of real numbers (a "real vector space") yields a vector space V C over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers.
The Orlando Magic will be without Franz Wagner indefinitely after the forward suffered a torn right oblique during Friday's loss to the Philadelphia 76ers, according to ESPN's Shams Charania ...
It contains calcium, magnesium, zinc and more — which health experts say means it's a nutritional option full of numerous benefits. "You're getting collagen; you get iron, ...
The bioctonion algebra is sometimes introduced as simply the complexification of real octonions, but in abstract algebra it is the result of the Cayley–Dickson construction that begins with the field of complex numbers, the trivial involution, and quadratic form z 2. The algebra of bioctonions is an example of an octonion algebra.
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