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In mathematics, the Smith–Minkowski–Siegel mass formula (or Minkowski–Siegel mass formula) is a formula for the sum of the weights of the lattices (quadratic forms) in a genus, weighted by the reciprocals of the orders of their automorphism groups. The mass formula is often given for integral quadratic forms, though it can be generalized ...
A fundamental pair of periods is a pair of complex numbers , such that their ratio / is not real. If considered as vectors in , the two are linearly independent.The lattice generated by and is
It is easy to show that the trace of a matrix representing an element of Γ(N) cannot be −1, 0, or 1, so these subgroups are torsion-free groups. (There are other torsion-free subgroups.) The principal congruence subgroup of level 2, Γ(2), is also called the modular group Λ. Since PSL(2, Z/2Z) is isomorphic to S 3, Λ is a subgroup of index 6.
In complex analysis, Gauss's continued fraction is a particular class of continued fractions derived from hypergeometric functions. It was one of the first analytic continued fractions known to mathematics, and it can be used to represent several important elementary functions , as well as some of the more complicated transcendental functions .
The continuant (,, …,) can be computed by taking the sum of all possible products of x 1,...,x n, in which any number of disjoint pairs of consecutive terms are deleted (Euler's rule).
As regards the short-rate models, these are, in turn, further categorized: these will be either equilibrium-based (Vasicek and CIR) or arbitrage-free (Ho–Lee and subsequent). This distinction: for equilibrium-based models the yield curve is an output from the model, while for arbitrage-free models the yield curve is an input to the model. [32]
The Ising model is given by the usual cubic lattice graph = (,) where is an infinite cubic lattice in or a period cubic lattice in , and is the edge set of nearest neighbours (the same letter is used for the energy functional but the different usages are distinguishable based on context).
The complex torus associated to a lattice spanned by two periods, ω 1 and ω 2.Corresponding edges are identified. In mathematics, a complex torus is a particular kind of complex manifold M whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number N circles).