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for which a ≡ d ≡ ±1 (mod N) and b ≡ c ≡ 0 (mod N). 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.
In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example is the subgroup of invertible 2 × 2 integer matrices of determinant 1 in which the off-diagonal entries are even.
The gamma function is an important special function in mathematics.Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general.
In mathematics, the p-adic gamma function Γ p is a function of a p-adic variable analogous to the gamma function. It was first explicitly defined by Morita (1975), though Boyarsky (1980) pointed out that Dwork (1964) implicitly used the same function. Diamond (1977) defined a p-adic analog G p of log Γ.
In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface viewed up to continuous (in the compact-open topology) deformation.
Teddi Mellencamp is using her own health journey to create awareness for World Cancer Day.. On Tuesday, Feb. 4, the Real Housewives of Beverly Hills alum, 43, re-posted a graphic photo of her ...
This article was reviewed by Craig Primack, MD, FACP, FAAP, FOMA. We’ve all been there. We grab an extra slice of pizza even though we’re full or reach for a snack when bored, tired, stressed ...
Since the gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entire function. As an entire function, it is of order 1 (meaning that log log | 1 / Γ( z ) | grows no faster than log | z | ), but of infinite type (meaning that log | 1 / Γ( z ) | grows faster than any multiple of | z ...