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In mathematics, the modular group is the projective special linear group (,) of matrices with integer coefficients and determinant, such that the matrices and are identified. The modular group acts on the upper-half of the complex plane by linear fractional transformations .
A modular function is a function that is invariant with respect to the modular group, but without the condition that it be holomorphic in the upper half-plane (among other requirements). Instead, modular functions are meromorphic: they are holomorphic on the complement of a set of isolated points, which are poles of the function.
In mathematics, a Hilbert modular form is a generalization of modular forms to functions of two or more variables. It is a (complex) analytic function on the m -fold product of upper half-planes H {\displaystyle {\mathcal {H}}} satisfying a certain kind of functional equation .
Simulink is a MATLAB-based graphical programming environment for modeling, simulating and analyzing multidomain dynamical systems. Its primary interface is a graphical block diagramming tool and a customizable set of block libraries .
The modular function is a continuous group homomorphism from G to the multiplicative group of positive real numbers. A group is called unimodular if the modular function is identically 1 {\displaystyle 1} , or, equivalently, if the Haar measure is both left and right invariant.
The modular group SL(2, Z) acts on the upper half-plane by fractional linear transformations.The analytic definition of a modular curve involves a choice of a congruence subgroup Γ of SL(2, Z), i.e. a subgroup containing the principal congruence subgroup of level N for some positive integer N, which is defined to be
It contains the modular group PSL(2, Z). Also closely related is the 2-fold covering group, Mp(2, R), a metaplectic group (thinking of SL(2, R) as a symplectic group). Another related group is SL ± (2, R), the group of real 2 × 2 matrices with determinant ±1; this is more commonly used in the context of the modular group, however.
It is not possible for a modular graph to contain a cycle of odd length. For, if C is a shortest odd cycle in a graph, x is a vertex of C, and yz is the edge of C farthest from x, there could be no median m(x, y, z). In this case, the only vertices on the shortest path yz are y and z themselves. Neither can belong to a shortest path from x to ...