21(2)
... If we substitute Z + (TT/2) for X in the trigonometric identity associated with (7), we find a formula whose associated hyperbolic one is -z - cosh 2x + cosh kx - cosh 6x +••• + (-l)^cosh 2kx ...
... If we substitute Z + (TT/2) for X in the trigonometric identity associated with (7), we find a formula whose associated hyperbolic one is -z - cosh 2x + cosh kx - cosh 6x +••• + (-l)^cosh 2kx ...
Document
... Case III: Q (x) contains irreducible quadratic factors, none of which is repeated. If Q (x) has the factor ax2 + bx + c, where b2 – 4ac < 0, then, in addition to the partial fractions, the expression for R (x)/Q (x) will have a term of the form ...
... Case III: Q (x) contains irreducible quadratic factors, none of which is repeated. If Q (x) has the factor ax2 + bx + c, where b2 – 4ac < 0, then, in addition to the partial fractions, the expression for R (x)/Q (x) will have a term of the form ...
definability of linear equation systems over
... forms for first-order logic extended with operators for solvability over finite fields. While it is known that solvability of linear equation systems over finite domains is not expressible in fixed-point logic with counting, it has also been observed that the logic can define many other natural prob ...
... forms for first-order logic extended with operators for solvability over finite fields. While it is known that solvability of linear equation systems over finite domains is not expressible in fixed-point logic with counting, it has also been observed that the logic can define many other natural prob ...
(pdf)
... Prime numbers are especially important for random number generators, making them useful in many algorithms. The Fermat Test uses Fermat’s Little Theorem to test for primality. Although the test is not guaranteed to work, it is still a useful starting point because of its simplicity and efficiency. A ...
... Prime numbers are especially important for random number generators, making them useful in many algorithms. The Fermat Test uses Fermat’s Little Theorem to test for primality. Although the test is not guaranteed to work, it is still a useful starting point because of its simplicity and efficiency. A ...
![9 The resultant and a modular gcd algorithm in Z[x]](http://s1.studyres.com/store/data/017337470_1-8c6dfa8fbd5c9da3a383209d818c2d7f-300x300.png)
HOMOMORPHISMS 1. Introduction
... Example 2.1. For each real number c, the formula c(x + y) = cx + cy for all x and y in R says that the function Mc : R → R where Mc (x) = cx is a group homomorphism. Example 2.2. For all real numbers x and y, |xy| = |x||y|. Therefore the absolute value function f : R× → R>0 , given by f (x) = |x|, i ...
... Example 2.1. For each real number c, the formula c(x + y) = cx + cy for all x and y in R says that the function Mc : R → R where Mc (x) = cx is a group homomorphism. Example 2.2. For all real numbers x and y, |xy| = |x||y|. Therefore the absolute value function f : R× → R>0 , given by f (x) = |x|, i ...
A Book of Abstract Algebra
... During the seven years that have elapsed since publication of the first edition of A Book of Abstract Algebra, I have received letters from many readers with comments and suggestions. Moreover, a number of reviewers have gone over the text with the aim of finding ways to increase it ...
... During the seven years that have elapsed since publication of the first edition of A Book of Abstract Algebra, I have received letters from many readers with comments and suggestions. Moreover, a number of reviewers have gone over the text with the aim of finding ways to increase it ...
Reasoning with Divisibility Mathematics Curriculum 4
... behaves differently than other numbers. Because of this difference, we don’t classify it as prime or composite. Many students might reason that 1 should be prime since its only factors are 1 and itself. In fact, their logic is sound, and throughout history, many mathematicians would have agreed! How ...
... behaves differently than other numbers. Because of this difference, we don’t classify it as prime or composite. Many students might reason that 1 should be prime since its only factors are 1 and itself. In fact, their logic is sound, and throughout history, many mathematicians would have agreed! How ...
Factorization
In mathematics, factorization (also factorisation in some forms of British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomial x2 − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained.The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viète's formulas relate the coefficients of a polynomial to its roots.The opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms.Integer factorization for large integers appears to be a difficult problem. There is no known method to carry it out quickly. Its complexity is the basis of the assumed security of some public key cryptography algorithms, such as RSA.A matrix can also be factorized into a product of matrices of special types, for an application in which that form is convenient. One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types: QR decomposition, LQ, QL, RQ, RZ.Another example is the factorization of a function as the composition of other functions having certain properties; for example, every function can be viewed as the composition of a surjective function with an injective function. This situation is generalized by factorization systems.