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In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer d such that the imaginary quadratic field [] has class number 1. Equivalently, the ring of algebraic integers of Q [ − d ] {\displaystyle \mathbb {Q} \left[{\sqrt {-d}}\right]} has unique factorization .
For a given cation, Pauling defined [2] the electrostatic bond strength to each coordinated anion as =, where z is the cation charge and ν is the cation coordination number. A stable ionic structure is arranged to preserve local electroneutrality , so that the sum of the strengths of the electrostatic bonds to an anion equals the charge on ...
In number theory, the Heegner theorem [1] establishes the complete list of the quadratic imaginary number fields whose rings of integers are principal ideal domains. It solves a special case of Gauss's class number problem of determining the number of imaginary quadratic fields that have a given fixed class number .
Kurt Heegner was a German mathematician; Heegner points are special points on elliptic curves; The Stark–Heegner theorem identifies the imaginary quadratic fields of class number 1. A Heegner number is a number n such that Q(√ −n) is an imaginary quadratic field of class number 1.
Solids with purely metallic bonding are characteristically ductile and, in their pure forms, have low strength; melting points can [inconsistent] be very low (e.g., Mercury melts at 234 K (−39 °C)). These properties are consequences of the non-directional and non-polar nature of metallic bonding, which allows atoms (and planes of atoms in a ...
The ring of formal power series over the complex numbers is a UFD, but the subring of those that converge everywhere, in other words the ring of entire functions in a single complex variable, is not a UFD, since there exist entire functions with an infinity of zeros, and thus an infinity of irreducible factors, while a UFD factorization must be ...
In mathematics, a Heegner point is a point on a modular curve that is the image of a quadratic imaginary point of the upper half-plane. They were defined by Bryan Birch and named after Kurt Heegner , who used similar ideas to prove Gauss's conjecture on imaginary quadratic fields of class number one.
The simultaneous generating series for the values on Heegner divisors and integrals along geodesic cycles of Klein's J-function (normalized such that the constant term vanishes) is a harmonic Maass form of weight 1/2.