4.1 Reduction theory
... Observe the minimum nonzero value of a reduced form R(x, y) is obtained precisely when (x, y) = (±1, 0), so we must have r = ±1, t = 0.† (Hence the minimum positive value of a reduced form is the x2 -coefficient, in this case a.) Further ru − st = ru = 1 implies u = r−1 . Then the xy-coefficient of S(x ...
... Observe the minimum nonzero value of a reduced form R(x, y) is obtained precisely when (x, y) = (±1, 0), so we must have r = ±1, t = 0.† (Hence the minimum positive value of a reduced form is the x2 -coefficient, in this case a.) Further ru − st = ru = 1 implies u = r−1 . Then the xy-coefficient of S(x ...
Degrees of curves in abelian varieties
... Let (X, A) be a principally polarized abelian variety of dimension n defined over an algebraically closed field k. The degree of a curve C contained in X is d = C • X. The first question we are interested in is to find what numbers can be degrees of irreducibles curves C. When C generates X, we prov ...
... Let (X, A) be a principally polarized abelian variety of dimension n defined over an algebraically closed field k. The degree of a curve C contained in X is d = C • X. The first question we are interested in is to find what numbers can be degrees of irreducibles curves C. When C generates X, we prov ...
Section 2.2
... Solution: Recall that for an affine cipher y (ax b) mod 26 , x is the numerical representation of the plaintext letter and y is the numerical representation of the ciphertext letter. Hence, using the MOD 26 alphabet assignment and the equation y (ax b) mod 26 , we see that Plaintext E x ...
... Solution: Recall that for an affine cipher y (ax b) mod 26 , x is the numerical representation of the plaintext letter and y is the numerical representation of the ciphertext letter. Hence, using the MOD 26 alphabet assignment and the equation y (ax b) mod 26 , we see that Plaintext E x ...
Number theory.pdf
... Unit 1: Properties of integers and linear Diophantine equations Level 1. Priority B. Basic Mathematics 2 is prerequisite. Properties of integers. Divisibility with remainder. Prime numbers and their distribution. Euclid’s proof of infinitely many primes. Euclid’s algorithm. Consequences, residue cla ...
... Unit 1: Properties of integers and linear Diophantine equations Level 1. Priority B. Basic Mathematics 2 is prerequisite. Properties of integers. Divisibility with remainder. Prime numbers and their distribution. Euclid’s proof of infinitely many primes. Euclid’s algorithm. Consequences, residue cla ...
Diophantine Equations CMT: 2011-2012
... vp (z) + vp (w) = vp (b), vp (x) + vp (z) = vp (c), and vp (y) + vp (w) = vp (d), where min(vp (y), vp (z)) = 0. Now it’s just a simple system of equations; see if you can finish the argument. There’s also another perspective to this problem. Rewrite the equation as a/c = d/b = y/z, where gcd(y, z) ...
... vp (z) + vp (w) = vp (b), vp (x) + vp (z) = vp (c), and vp (y) + vp (w) = vp (d), where min(vp (y), vp (z)) = 0. Now it’s just a simple system of equations; see if you can finish the argument. There’s also another perspective to this problem. Rewrite the equation as a/c = d/b = y/z, where gcd(y, z) ...
On the existence of equiangular tight frames
... There are two families of ETFs that arise for every dimension d and one family in dimension one with arbitrary number of vectors. (1) (Orthonormal Bases). When N = d, the sole examples of ETFs are unitary (and orthogonal) matrices. Evidently, the absolute inner product α between distinct vectors is ...
... There are two families of ETFs that arise for every dimension d and one family in dimension one with arbitrary number of vectors. (1) (Orthonormal Bases). When N = d, the sole examples of ETFs are unitary (and orthogonal) matrices. Evidently, the absolute inner product α between distinct vectors is ...
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.