![arXiv:math/0607274v2 [math.GT] 21 Jun 2007](http://s1.studyres.com/store/data/004624723_1-689e7c37d19114c6d0cb2b7e64bf5915-300x300.png)
arXiv:math/0607274v2 [math.GT] 21 Jun 2007
... (C \ {n points}) × C, so has the homotopy type of a bouquet of n circles. On the other hand, CP2 \ N = (D2 \ {n disks}) × D2 ; hence M is diffeomorphic to the n-fold connected sum ♯n S 1 × S 2 . Example 2.3 Let A be a near-pencil of n + 1 lines in CP2 , defined by the polynomial Q = x0 (xn1 − xn2 ). ...
... (C \ {n points}) × C, so has the homotopy type of a bouquet of n circles. On the other hand, CP2 \ N = (D2 \ {n disks}) × D2 ; hence M is diffeomorphic to the n-fold connected sum ♯n S 1 × S 2 . Example 2.3 Let A be a near-pencil of n + 1 lines in CP2 , defined by the polynomial Q = x0 (xn1 − xn2 ). ...
Fraction
... We’ll use the GCF method first, and then the Prime Factorization method. GCF Method for Reducing Step 1: Find the GCF of the numerator and denominator Step 2: Factor the numerator and denominator using GCF Step 3: Cancel the GCF using the fact that a number over itself is always 1 Step 3: Rewrite th ...
... We’ll use the GCF method first, and then the Prime Factorization method. GCF Method for Reducing Step 1: Find the GCF of the numerator and denominator Step 2: Factor the numerator and denominator using GCF Step 3: Cancel the GCF using the fact that a number over itself is always 1 Step 3: Rewrite th ...
Solution Set 1 - Williams College
... Write 7 = ab , where we may safely assume that ab is a reduced fraction (i.e., that a and b are both positive integers with no common factor greater than 1; if not, reduce the fraction!). It follows that a2 = 7b2 , ...
... Write 7 = ab , where we may safely assume that ab is a reduced fraction (i.e., that a and b are both positive integers with no common factor greater than 1; if not, reduce the fraction!). It follows that a2 = 7b2 , ...
Complex number
... introduce it here. Briefly, complex numbers are added, subtracted, and multiplied by formally applying the associative, commutative and distributive laws of algebra. The set of complex numbers forms a field which, in contrast to the real numbers, is algebraically closed. In mathematics, the adjectiv ...
... introduce it here. Briefly, complex numbers are added, subtracted, and multiplied by formally applying the associative, commutative and distributive laws of algebra. The set of complex numbers forms a field which, in contrast to the real numbers, is algebraically closed. In mathematics, the adjectiv ...
Sum of Cubes
... So now, let’s ask about column two. To take an example, consider the divisor 12 of 72. I ask how many divisors 12 has, but I am going to try to find the answer, not by looking in column two in table 72 (where there’s a 6 sitting right beside it) but by trekking over to tables 8 and 9. To do that, I ...
... So now, let’s ask about column two. To take an example, consider the divisor 12 of 72. I ask how many divisors 12 has, but I am going to try to find the answer, not by looking in column two in table 72 (where there’s a 6 sitting right beside it) but by trekking over to tables 8 and 9. To do that, I ...
QUATERNION ALGEBRAS 1. Introduction = −1. Addition and multiplication
... the distributive law, to multiply any two quaternions. Example 2.2. (i + j)(i − j) = i2 − ij + ji − j 2 = −1 − k − k − (−1) = −2k, while i2 − j 2 = −1 − (−1) = 0. Example 2.3. A quaternion with a = 0 is called a pure quaternion, and the square of a pure quaternion is a negative sum of three squares: ...
... the distributive law, to multiply any two quaternions. Example 2.2. (i + j)(i − j) = i2 − ij + ji − j 2 = −1 − k − k − (−1) = −2k, while i2 − j 2 = −1 − (−1) = 0. Example 2.3. A quaternion with a = 0 is called a pure quaternion, and the square of a pure quaternion is a negative sum of three squares: ...
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.