NAVAL POSTGRADUATE SCHOOL
... and we explore the set of polynomials x 2 + x + α j over GF(2n) where α is a primitive element of GF(2n). In particular, we look at the structure of the constant coefficients of the polynomials. A primitive element of a Galois field of size pn is an element whose powers are all different. Since ther ...
... and we explore the set of polynomials x 2 + x + α j over GF(2n) where α is a primitive element of GF(2n). In particular, we look at the structure of the constant coefficients of the polynomials. A primitive element of a Galois field of size pn is an element whose powers are all different. Since ther ...
31(2)
... When t(T) < 1, regardless of the value of Pi(T), equation (1) shows that equality holds on the right in the n odd cases of Figure 1 and on the left in the n even cases. Other trees can be obtained by appending an arbitrary number of degree-one neighbors to the degree-two nodes in either of the trees ...
... When t(T) < 1, regardless of the value of Pi(T), equation (1) shows that equality holds on the right in the n odd cases of Figure 1 and on the left in the n even cases. Other trees can be obtained by appending an arbitrary number of degree-one neighbors to the degree-two nodes in either of the trees ...
36(4)
... The main tool used in proving this theorem is a certain generalization of the famous averagetheorem of Gauss-Kusmin-Levy concerning the elements of continued fractions (see Satz 35 in [4]), which is stated in Lemma 2.1 below. It follows from [5] or [7]. The set si given in Theorem 1.1 depends on s a ...
... The main tool used in proving this theorem is a certain generalization of the famous averagetheorem of Gauss-Kusmin-Levy concerning the elements of continued fractions (see Satz 35 in [4]), which is stated in Lemma 2.1 below. It follows from [5] or [7]. The set si given in Theorem 1.1 depends on s a ...
Homework for Maths For Classes VII to XII
... 1. The radius of a circle is increasing uniformly at the rate of 3cm/sec. Find the rate at which the area of circle is increasing when the radius is 10cm. 2. A balloon which always spherical, has a variable diameter 3/2(2x+1). Find the rate of change of its volume with respect to x. 3. The total rev ...
... 1. The radius of a circle is increasing uniformly at the rate of 3cm/sec. Find the rate at which the area of circle is increasing when the radius is 10cm. 2. A balloon which always spherical, has a variable diameter 3/2(2x+1). Find the rate of change of its volume with respect to x. 3. The total rev ...
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.