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LOWER BOUNDS FOR Z-NUMBERS 1. An approximate
LOWER BOUNDS FOR Z-NUMBERS 1. An approximate

Aalborg Universitet Real-Time Implementations of Sparse Linear Prediction for Speech Processing
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... There are many methods for solving the sparse LPC (5). In this paper, we will focus on primal-dual interior-point methods. The purpose is twofold. Firstly, these are well known efficient methods for convex optimization used in real-time convex optimization [19, 20]; Secondly, by comparing the propos ...
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... results that only one fundamental solution (x, y, z) exists for p and q (one of which is odd and the other even). Based on 8, the previous definition is re-defined to: for any numbers p and q (one of which is odd and the other even) there are at least two fundamental solutions (x1 , y1 , z1 ) and (x2 ...
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on strings of consecutive integers with no large prime factors

... Our most general conclusions are somewhat complicated to state, and hence we defer their enunciation to Section 2 below. At this point we content ourselves by recording two direct consequences of our methods. Our first result shows that there are arbitrarily long strings of consecutive smooth number ...
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... without row or column exchanges to give its unique solution and the computations are stable with respect to roundo error. De nition of Positive De nite Matrix. A is positive de nite if it is symmetric and if xt Ax > 0 for every x 6= 0. Theorem of Positive De nite Matrix If A is n  n positive de ni ...
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Holt McDougal Algebra 2 3-5

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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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