Quasi-random numbers in stochastic finite element analysis
... Uncertainty propagation - stochastic finite elements - polynomial chaos expansion - quasirandom numbers - structural reliability 1 Introduction Computer simulations are nowadays commonly used in structural engineering to accurately model the behaviour of complex systems. Most of them are determinist ...
... Uncertainty propagation - stochastic finite elements - polynomial chaos expansion - quasirandom numbers - structural reliability 1 Introduction Computer simulations are nowadays commonly used in structural engineering to accurately model the behaviour of complex systems. Most of them are determinist ...
On a different kind of d -orthogonal polynomials that generalize the Laguerre polynomials
... with the regularity conditions ad+1,d (n) a0,d (n) 6= 0, n ≥ d, and under the convention P−n = 0, n ≥ 1. For the particular case d = 1, this theorem is reduced to the so-called Favard Theorem [16]. The d-orthogonality notion seems to appear in various domains of mathematics. For instance, there is a ...
... with the regularity conditions ad+1,d (n) a0,d (n) 6= 0, n ≥ d, and under the convention P−n = 0, n ≥ 1. For the particular case d = 1, this theorem is reduced to the so-called Favard Theorem [16]. The d-orthogonality notion seems to appear in various domains of mathematics. For instance, there is a ...
Notes: Quadratic Functions Vertex Form
... Writing Equations given the vertex. Steps: 1. Plug in __________________________ and the given _____________________________. 2. Solve for _____________. 3. Go back to _______________ _______________, plug in the ______________ and ______ value. ...
... Writing Equations given the vertex. Steps: 1. Plug in __________________________ and the given _____________________________. 2. Solve for _____________. 3. Go back to _______________ _______________, plug in the ______________ and ______ value. ...
Full text
... Theorem 1.1: Let n be a nonsimple integer. Then n is semisimple if and only if n is a prime, or n = 32 , or n = aq1 · · · qk Ai with a(q1 − pi+1 ) · · · (qk − pi+1 ) < pi+1 , where i ≥ 1, k ≥ 0, q1 > q2 > · · · > qk > pi+1 are primes and a is a positive integer. In [3] it also asked for the set S + ...
... Theorem 1.1: Let n be a nonsimple integer. Then n is semisimple if and only if n is a prime, or n = 32 , or n = aq1 · · · qk Ai with a(q1 − pi+1 ) · · · (qk − pi+1 ) < pi+1 , where i ≥ 1, k ≥ 0, q1 > q2 > · · · > qk > pi+1 are primes and a is a positive integer. In [3] it also asked for the set S + ...
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.