
09-14-2011 1 Garrett Continuing the review of the simple (!?) case of number...
... with finite LHS, and infinite RHS... and noted that ideas from complex variables and Fourier analysis are needed to make this legitimate. A similar discussion applies to many other zeta functions and L-functions, such as those used by Dirichlet to prove the primes-in-arithmetic progressions theorem. ...
... with finite LHS, and infinite RHS... and noted that ideas from complex variables and Fourier analysis are needed to make this legitimate. A similar discussion applies to many other zeta functions and L-functions, such as those used by Dirichlet to prove the primes-in-arithmetic progressions theorem. ...
Script 2013W 104.271 Discrete Mathematics VO (Gittenberger)
... Kruskal’s and Prim’s algorithm are greedy algorithms. These algorithms only work on a local view of the graph and they use these local values to solve the maximization (or minimization) problem. Since these algorithms are greedy, they generally don’t produce optimal maximal or minimal spanning trees ...
... Kruskal’s and Prim’s algorithm are greedy algorithms. These algorithms only work on a local view of the graph and they use these local values to solve the maximization (or minimization) problem. Since these algorithms are greedy, they generally don’t produce optimal maximal or minimal spanning trees ...
supplemental sheet #5
... Rationalizing the denominator is an important idea as it simplifies algebraic manipulation, for example, in combining two fractions involving surds. Simplify the following fractions: ...
... Rationalizing the denominator is an important idea as it simplifies algebraic manipulation, for example, in combining two fractions involving surds. Simplify the following fractions: ...
Quaternion algebras and quadratic forms
... Proof. Note ↵ = tr↵ ↵. Since is an isomorphism, ↵ and (↵) have the same reduced characteristic polynomials. (Either ↵ 2 F and p↵ = (x ↵)2 , or ↵ 62 F and the characteristic polynomial is the same as the minimal polynomial.) In particular tr (↵) = tr↵. Thus ...
... Proof. Note ↵ = tr↵ ↵. Since is an isomorphism, ↵ and (↵) have the same reduced characteristic polynomials. (Either ↵ 2 F and p↵ = (x ↵)2 , or ↵ 62 F and the characteristic polynomial is the same as the minimal polynomial.) In particular tr (↵) = tr↵. Thus ...
Clock-Controlled Shift Registers for Key
... In order to ensure security of a key-stream generator against the BerlekampMassey algorithm, its output sequence should have large period and high linear complexity. On the other hand, good statistical properties of the output sequence prevent the reconstruction of statistically redundant plaintext ...
... In order to ensure security of a key-stream generator against the BerlekampMassey algorithm, its output sequence should have large period and high linear complexity. On the other hand, good statistical properties of the output sequence prevent the reconstruction of statistically redundant plaintext ...
Section Outlines - Handouts - University of Nebraska–Lincoln
... No: 5|(10d + 25q) but 5 does not divide 206 2) Given 100 coins consisting of pennies, dimes and quarters, can their value total $5.00? No: Subtract p+d+q=100 from p+10d+25q=500. The integer 3 divides the left hand side of the resulting equation but not the right hand side. Definition: greatest commo ...
... No: 5|(10d + 25q) but 5 does not divide 206 2) Given 100 coins consisting of pennies, dimes and quarters, can their value total $5.00? No: Subtract p+d+q=100 from p+10d+25q=500. The integer 3 divides the left hand side of the resulting equation but not the right hand side. Definition: greatest commo ...
20(3)
... The principal purpose of The Fibonacci Quarterly is to serve as a focal point for widespread interest in the Fibonacci and related numbers, especially with respect to new results, research proposals, and challenging problems. The Quarterly seeks articles that are intelligible yet stimulating to its ...
... The principal purpose of The Fibonacci Quarterly is to serve as a focal point for widespread interest in the Fibonacci and related numbers, especially with respect to new results, research proposals, and challenging problems. The Quarterly seeks articles that are intelligible yet stimulating to its ...
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.