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The Goldston-Pintz-Yıldırım sieve and some applications
The Goldston-Pintz-Yıldırım sieve and some applications

Families of ordinary abelian varieties
Families of ordinary abelian varieties

... that every Tate-linear subvariety of the ordinary locus of a Hilbert modular variety, attached to a totally real number field, is the reduction of a Shimura subvariety. Our argument can be used to establish the other conjectures in the special case of subvarieties of a Hilbert modular variety, altho ...
Here
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Modular functions and modular forms
Modular functions and modular forms

Centralized Management and Processing Policy for Log Files
Centralized Management and Processing Policy for Log Files

... stored. In order to do well to the data stored in the log files, one should do the three jobs as below. 5.1 To filtrate and extract The large amount of log data always confuse the administrator, so we should filter the large amount of log data, and then extract the log data worthy of being analyzed ...
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public_key_cryptography

CSCI 190 Additional Final Practice Problems Solutions
CSCI 190 Additional Final Practice Problems Solutions

Querying Large Collections of Semistructured Data
Querying Large Collections of Semistructured Data

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Chapter 3_Old

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Commutative Algebra
Commutative Algebra

Elements of Modern Algebra
Elements of Modern Algebra

Commutative Algebra
Commutative Algebra

Elliptic spectra, the Witten genus, and the theorem of the cube
Elliptic spectra, the Witten genus, and the theorem of the cube

Math 373 Exam 1 Instructions In this exam, Z denotes the set of all
Math 373 Exam 1 Instructions In this exam, Z denotes the set of all

On the Amount of Sieving in Factorization Methods
On the Amount of Sieving in Factorization Methods

NOETHERIAN MODULES 1. Introduction In a finite
NOETHERIAN MODULES 1. Introduction In a finite

´Etale cohomology of schemes and analytic spaces
´Etale cohomology of schemes and analytic spaces

Elliptic Curves with Complex Multiplication and the Conjecture of
Elliptic Curves with Complex Multiplication and the Conjecture of

... clearly a left-inverse of the first, and it maps into p by Lemma 3.5. We only need show that the second map is also one-to-one. If we rewrite our Weierstrass equation for E with variables w = −1/y and z = −x/y we get a new equation a6 w3 + (a4 z + a3 )w2 + (a2 z 2 + a1 z − 1)w + z 3 = 0. Fix a value ...
PDF of Version 2.01-B of GIAA here.
PDF of Version 2.01-B of GIAA here.

Lectures on Hopf algebras
Lectures on Hopf algebras

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A. Powers of 10

A Comparative Analysis of Different Categorical Data Clustering
A Comparative Analysis of Different Categorical Data Clustering

Moduli of elliptic curves
Moduli of elliptic curves

SECTION C Solving Linear Congruences
SECTION C Solving Linear Congruences

... Solving this equation gives x  3 . A linear congruence is an equation of the form ax  b  mod n  The solution of this linear congruence is the set of integers x which satisfies this: ax  b  mod n  Why is the solution a set of integers rather than a unique integer? Remember ax  b  mod n  mea ...
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Factorization of polynomials over finite fields

In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to allow the computation of the factorization by means of an algorithm. In practice, algorithms have been designed only for polynomials with coefficients in a finite field, in the field of rationals or in a finitely generated field extension of one of them.The case of the factorization of univariate polynomials over a finite field, which is the subject of this article, is especially important, because all the algorithms (including the case of multivariate polynomials over the rational numbers), which are sufficiently efficient to be implemented, reduce the problem to this case (see Polynomial factorization). It is also interesting for various applications of finite fields, such as coding theory (cyclic redundancy codes and BCH codes), cryptography (public key cryptography by the means of elliptic curves), and computational number theory.As the reduction of the factorization of multivariate polynomials to that of univariate polynomials does not have any specificity in the case of coefficients in a finite field, only polynomials with one variable are considered in this article.
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