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That implies that any two rational functions F and G, in the function field of the modular curve, will satisfy a modular equation P(F,G) = 0 with P a non-zero polynomial of two variables over the complex numbers. For suitable non-degenerate choice of F and G, the equation P(X,Y) = 0 will actually define the modular curve.
Konopasek's goal in inventing the TK Solver concept was to create a problem solving environment in which a given mathematical model built to solve a specific problem could be used to solve related problems (with a redistribution of input and output variables) with minimal or no additional programming required: once a user enters an equation, TK ...
The matrices [e 1, ..., e n] are divisible by all non-zero linear forms in the variables X i with coefficients in the finite field F q. In particular the Moore determinant [0, 1, ..., n − 1] is a product of such linear forms, taken over 1 + q + q 2 + ... + q n – 1 representatives of ( n – 1)-dimensional projective space over the field.
Numerical analysis is an area of mathematics that creates and analyzes algorithms for obtaining numerical approximations to problems involving continuous variables. When an arbitrary function does not have a closed form as its solution, there would not be any analytical tools present to evaluate the desired solutions, hence an approximation ...
In mathematics, Felix Klein's j-invariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for special linear group SL(2, Z) defined on the upper half-plane of complex numbers. It is the unique such function that is holomorphic away from a simple pole at the cusp such that
The General Problem Solver (GPS) is a particular computer program created in 1957 by Herbert Simon, J. C. Shaw, and Allen Newell intended to work as a universal problem solver, that theoretically can be used to solve every possible problem that can be formalized in a symbolic system, given the right input configuration.
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.
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.