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In statistics, and particularly in econometrics, the reduced form of a system of equations is the result of solving the system for the endogenous variables. This gives the latter as functions of the exogenous variables, if any.
In linear algebra, reduction refers to applying simple rules to a series of equations or matrices to change them into a simpler form. In the case of matrices, the process involves manipulating either the rows or the columns of the matrix and so is usually referred to as row-reduction or column-reduction, respectively.
Since modular forms were reduced using the -series, it makes sense to use the -series definition. The sum simplifies a lot for Hecke operators of primes (i.e. when is prime): there are only two summands. This is very nice for reduction modulo 2, as the formula simplifies a lot.
The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...
In a weak formulation, equations or conditions are no longer required to hold absolutely (and this is not even well defined) and has instead weak solutions only with respect to certain "test vectors" or "test functions". In a strong formulation, the solution space is constructed such that these equations or conditions are already fulfilled.
A normal form for a group G with generating set S is a choice of one reduced word in S for each element of G. For example: For example: The words 1, i , j , ij are a normal form for the Klein four-group with S = { i , j } and 1 representing the empty word (the identity element for the group).
In the twentieth century work by Axel Thue, [6] Carl Siegel, [7] and Klaus Roth [8] reduced the exponent in Liouville's work from d + ε to d/2 + 1 + ε, and finally, in 1955, to 2 + ε. This result, known as the Thue–Siegel–Roth theorem , is ostensibly the best possible, since if the exponent 2 + ε is replaced by just 2 then the result is ...
The identity element of this algebra is the condensed interval [1, 1]. If interval [x, y] is not in one of the ideals, then it has multiplicative inverse [1/x, 1/y]. Endowed with the usual topology, the algebra of intervals forms a topological ring. The group of units of this ring consists of four quadrants determined by the axes, or ideals in ...
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