1 - UCLA Computer Science
... However, the derivation is beyond the scope of this paper, Moreover, it would be more complicated and difficult to understand than the simple visual proof presented above. Conclusions We have demonstrated simple visual proofs that for any integer n 2, n3 can be represented as the sum of consecuti ...
... However, the derivation is beyond the scope of this paper, Moreover, it would be more complicated and difficult to understand than the simple visual proof presented above. Conclusions We have demonstrated simple visual proofs that for any integer n 2, n3 can be represented as the sum of consecuti ...
Greatest Common Factor The greatest common factor of two or more
... Adding and Subtracting Mixed Numbers To add or subtract mixed numbers, simply convert the mixed numbers into improper fractions, then add or subtract them as fractions. Example: ...
... Adding and Subtracting Mixed Numbers To add or subtract mixed numbers, simply convert the mixed numbers into improper fractions, then add or subtract them as fractions. Example: ...
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... tiling of length k − 1. We would then look for a tiling of length k − 2, starting with tiling T2 . Otherwise, T1 is breakable at cell k − 2, followed by a domino (which happens fk−2 fn−k−1 ways. Here, we “throw away” cells 1 through k, and consider the remaining cells to be a new tiling, which we ca ...
... tiling of length k − 1. We would then look for a tiling of length k − 2, starting with tiling T2 . Otherwise, T1 is breakable at cell k − 2, followed by a domino (which happens fk−2 fn−k−1 ways. Here, we “throw away” cells 1 through k, and consider the remaining cells to be a new tiling, which we ca ...
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.