Algebraic Statistics
... The emerging field of algebraic statistics advocates the use of polynomial algebra as a tool for statistical analysis. The underlying principle is that many natural families of probability distributions on discrete random variables are algebraic varieties (the zero sets of polynomials). Knowing the ...
... The emerging field of algebraic statistics advocates the use of polynomial algebra as a tool for statistical analysis. The underlying principle is that many natural families of probability distributions on discrete random variables are algebraic varieties (the zero sets of polynomials). Knowing the ...
Document
... For any number a, 1 a a and a 1 a . (Multiplying a number by 1 gives that number.) ...
... For any number a, 1 a a and a 1 a . (Multiplying a number by 1 gives that number.) ...
February 23
... (x+y+z)^n = sum (n \multichoose a,b,c) x^a y^b z^c, where the summation extends over all non-negative integers a,b,c with a+b+c=n. Section 5.6: Newton's binomial theorem Note that (n \choose 2) = n(n-1)/2, and that this makes sense even when n is not an integer. More generally, one can define (r \c ...
... (x+y+z)^n = sum (n \multichoose a,b,c) x^a y^b z^c, where the summation extends over all non-negative integers a,b,c with a+b+c=n. Section 5.6: Newton's binomial theorem Note that (n \choose 2) = n(n-1)/2, and that this makes sense even when n is not an integer. More generally, one can define (r \c ...
Definition Sheet
... the form ax^n where a is the coefficient and n is the exponent.” When writing a polynomial you want to put it into standard form where the expressions are written in descending order from the largest degree to the smallest degree, which is what we have done. A degree is the exponent of the variable ...
... the form ax^n where a is the coefficient and n is the exponent.” When writing a polynomial you want to put it into standard form where the expressions are written in descending order from the largest degree to the smallest degree, which is what we have done. A degree is the exponent of the variable ...
Name: Date: Period:______ Unit 1 Practice Test Variables and
... Variables and Translations. Write an algebraic expression for each phrase. a. the sum of 5 and three times a number _________________________________________________ b. 8 minus the product of 9 and a number _______________________________________ Define variables and write an equation to model the f ...
... Variables and Translations. Write an algebraic expression for each phrase. a. the sum of 5 and three times a number _________________________________________________ b. 8 minus the product of 9 and a number _______________________________________ Define variables and write an equation to model the f ...
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.