In the sequel we assume that (rhoinRe) is a convex, σ-finite modular function, and C is a nonempty subset of the modular function space (L_{rho}).
It is easy to check that ρ is a convex modular function and the corresponding modular space is called the Musielak-Orlicz space and is denoted by L φ.
Then b, being the sum of a modular function and a submodular function ((b(X)=frac{1}{2}sum_{vin X}(d^_{A_{1}} v -d^_{A_{1} v -d^_{E}(v))+d^_{A_{0}}(X))), is a submodular function and, by the assumption, it is integer valued.
The Moonshine conjectures asserted a mysterious connection between certain families of modular functions and the representation theory of the largest sporadic simple group (the "Monster").
A similar theory can be developed for special values of elliptic or elliptic modular functions, and is called complex multiplication of such functions.
The proofs of all these identities and congruences heavily depend upon the theory of modular functions and the properties of Eisenstein series.
Corollary 3.6 Let ( X, ⪯, ρ ) be a complete ordered modular function space and S and T, continuous self-maps on X ρ, ( S, T ) and ( T, S ) are ρ-partially weakly increasing with respect to identity mapping on X, and ρ ( S f − T g ) ≤ α ρ ( f − g ).
Corollary 3.7 Let ( X, ⪯, ρ ) be a complete ordered modular function space and S and J continuous self-maps on X ρ.
Corollary 3.2 Let ( X, ⪯, ρ ) be a complete ordered modular function space and S, I, and J self-maps on X such that S ( X ) ⊆ J X X ), and I ( X ) ⊆ S ( X ).
Corollary 3.3 Let ( X, ⪯, ρ ) be a complete ordered modular function space and S, T, and J self-maps on X such that S ( L ρ ) ⊆ J ( L ρ ) and J ( L ρ ) ⊆ T ( L ρ ).
Theorem 3.5 Let ( X, ⪯, ρ ) be a complete ordered modular function space and S, T, I, and J continuous self-maps on X ρ.
