Miscellaneous Problems Index
... 4.8 To: Second-order recursion relation 7.2(1969)200 4.9 To: Second-order recursion relation 7.2(1969)200 4.10 To: Recursion from a Binet-type relation 7.2(1969)200 5.1 To: Recursion relation for a given sequence 7.3(1969)300 [The answers to problems 1 - 5 are in 7.2(1969)210 and 6-10 are in 7.2(196 ...
... 4.8 To: Second-order recursion relation 7.2(1969)200 4.9 To: Second-order recursion relation 7.2(1969)200 4.10 To: Recursion from a Binet-type relation 7.2(1969)200 5.1 To: Recursion relation for a given sequence 7.3(1969)300 [The answers to problems 1 - 5 are in 7.2(1969)210 and 6-10 are in 7.2(196 ...
Interesting problems from the AMATYC Student Math League Exams
... So the correct answer is D) 96. [See the section on Logarithmic Properties] (February 2004, #7) The number 877530p765q6 is divisible by both 8 and 11, with p and q both digits from 0 to 9. The number is also divisible by In order for the number to be divisible by 8, 5q6, the number consisting of the ...
... So the correct answer is D) 96. [See the section on Logarithmic Properties] (February 2004, #7) The number 877530p765q6 is divisible by both 8 and 11, with p and q both digits from 0 to 9. The number is also divisible by In order for the number to be divisible by 8, 5q6, the number consisting of the ...
The Impact of Disjunction on Query Answering Under Guarded-Based Existential Rules
... years, see, e.g., [1–5]. This interest stems from the inability of plain Datalog to infer the existence of new objects which are not already in the extensional database [6]. The obtained rules are known under a variety of names such as existential rules, tuplegenerating dependencies (TGDs), and Data ...
... years, see, e.g., [1–5]. This interest stems from the inability of plain Datalog to infer the existence of new objects which are not already in the extensional database [6]. The obtained rules are known under a variety of names such as existential rules, tuplegenerating dependencies (TGDs), and Data ...
2005 Exam
... 20. A 20-foot by 30-foot rectangular barn sits in the middle of a flat, open field. The farmer wants to tether a goat to the barn using a chain 50 feet long. The goat cannot go under, into, or through the barn. If the farmer wishes to provide the goat with the maximum possible grazing area, then th ...
... 20. A 20-foot by 30-foot rectangular barn sits in the middle of a flat, open field. The farmer wants to tether a goat to the barn using a chain 50 feet long. The goat cannot go under, into, or through the barn. If the farmer wishes to provide the goat with the maximum possible grazing area, then th ...
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.