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Rational Functions and its geometric Notions

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A function is called a rational function if and only if it can be written in the form
where and are polynomials in and is not the zero polynomial. The domain of is the set of all points for which the denominator is not zero, assuming and have no common factors.

In abstract algebra the concept of a polynomial is extended to include formal expressions in which the coefficients of the polynomial can be taken from any field. In this setting given a field F and some indeterminate X, a rational expression is any element of the field of fractions of the polynomial ring F[X]. Any rational expression can be written as the quotient of two polynomials P/Q with Q ≠ 0, although this representation isn't unique. P/Q is equivalent to R/S, for polynomials P, Q, R, and S, when PS = QR. However since F[X] is a unique factorization domain, there is a unique representation for any rational expression P/Q with P and Q polynomials of lowest degree and Q chosen to be monic. This is similar …

Basics Of Set Theory

Set theory is the branch of mathematical logic that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.
 Basic concepts and notation Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member (or element) of A, write o ∈ A. Since sets are objects, the membership relation can relate sets as well.
A derived binary relation between two sets is the subset relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, denoted A ⊆ B. For example, {1,2} is a subset of {1,2,3} , but {1,4} is not. From this definition, it is clear that a set is a subset of itself; for cases where one wishes to rule out this, the term proper subset is defined. A is called a proper subset o…

Group Algebras of Topological Groups

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In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra (or more generally a Banach algebra), such that representations of the algebra are related to representations of the group. As such, they are similar to the group ring associated to a discrete group.



Group algebras of topological groups: Cc(G) For the purposes of functional analysis, and in particular of harmonic analysis, one wishes to carry over the group ring construction to topological groups G. In case G is a locally compact Hausdorff group, G carries an essentially unique left-invariant countably additive Borel measure μ called Haar measure. Using the Haar measure, one can define a convolution operation on the space Cc(G) of complex-valued continuous functions on G with compact support; Cc(G) can then be given any of various norms and the completion will be a group algebra.
To define the convolution operation, let f and g be two functions in Cc(G). For t in …

Something About Factorization of Polynomials

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In mathematics and computer algebra the factorization of a polynomial consists in decomposing it in a product of irreducible factors. This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to allow to compute this factorization by mean of an algorithm. In practice, algorithms have been designed only for polynomials with coefficients in a finite field, in the field of rationals or in a finitely generated field extension of one of them.
The case of the factorization of univariate polynomials over a finite field, which is the object of this article, is especially important, because all the algorithms (including the case of multivariate polynomials over the rational numbers), which are sufficiently efficient to be implemented, reduce the problem to this case (see Polynomial factorization). It is also interesting for various applications of finite fields su…

Identity Without Variables in Trigonometory

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Identities without variables The curious identity
is a special case of an identity that contains one variable:
Similarly:
The same cosine identity in radians is
Similarly:
The following is perhaps not as readily generalized to an identity containing variables (but see explanation below):
Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:
The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than 21/2 that are relatively prime to (or have no prime factors in common with) 21. The last several examples are corollaries of a basic fact about the irreducible cyclotomic polynomials: the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the Möbius function evaluated at (in the very last case above) 21; only half of the zeroes are present above. The two identities preceding this last one arise in the same fashion with 21 replaced by 10 and…

Square Matrices in Mathematics

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Square matrices A square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied. The entries aii form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix.

Main typesNameExample with n = 3Diagonal matrixLower triangular matrixUpper triangular matrixDiagonal or triangular matrix If all entries outside the main diagonal are zero, A is called a diagonal matrix. If only all entries above (or below) the main diagonal are zero, A is called a lower (or upper) triangular matrix.
Identity matrix The identity matrix In of size n is the n-by-n matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, e.g.
It is a square matrix of order n, and also a special kind of diagonal matrix. It is called identity matrix because multip…