In order to enlarge the class of equations provided by traditional polynomials over a binary algebra A to a more useful class of equations, we introduce polynomials in noncommuting or nonassociating indeterminates. We discuss algebraic properties of these formal polynomial algebras and their accompanying polynomial function algebras. We present certain basis results for polynomial algebras, which are used to address the question of zero divisors in a polynomial algebra. We give an analog of the remainder theorem and the factor theorem for polynomials. Particular emphasis is placed on showing the difference between polynomials and polynomial functions. We also provide a brief discussion of polynomial composition and formal derivatives.

We present a faster implementation of the polynomial time primal simplex algorithm due to Orlin [23]. His algorithm requires O(nm min{log(nC), m log n}) pivots and O(n2 m ??n{log nC, m log n}) time. The bottleneck operations in his algorithm are performing the relabeling operations on nodes, selecting entering arcs for pivots, and performing the pivots. We show how to speed up these operations so as to yield an algorithm whose running time is O(nm. log n) per scaling phase. We show how to extend the dynamic-tree data-structure in order to implement these algorithms. The extension may possibly have other applications as well.

The multivariable Alexander polynomial (MVA) is a classical invariant of knots and links. We give an extension to regular virtual knots which has simple versions of many of the relations known to hold for the classical invariant. By following the previous proofs that the MVA is of finite type we give a new definition for its weight system which can be computed as the determinant of a matrix created from local information. This is an improvement on previous definitions as it is directly computable (not defined recursively) and is computable in polynomial time. We also show that our extension to virtual knots is a finite type invariant of virtual knots. We further explore how the multivariable Alexander polynomial takes local information and packages it together to form a global knot invariant, which leads us to an extension to tangles. To define this invariant we use so-called circuit algebras, an extension of planar algebras which are the `right' setting to discuss virtual knots. Our tangle invariant is a circuit algebra morphism, and so behaves well under tangle operations and gives yet another definition for the Alexander polynomial. The MVA and the single variable Alexander polynomial are known to satisfy a number of relations, each of which has a proof relying on different approaches and techniques. Using our invariant we can give simple computational proofs of many of these relations, as well as an alternate proof that the MVA and our virtual extension are of finite type.

The multivariable Alexander polynomial (MVA) is a classical invariant of knots and links. We give an extension to regular virtual knots which has simple versions of many of the relations known to hold for the classical invariant. By following the previous proofs that the MVA is of finite type we give a new definition for its weight system which can be computed as the determinant of a matrix created from local information. This is an improvement on previous definitions as it is directly computable (not defined recursively) and is computable in polynomial time. We also show that our extension to virtual knots is a finite type invariant of virtual knots. We further explore how the multivariable Alexander polynomial takes local information and packages it together to form a global knot invariant, which leads us to an extension to tangles. To define this invariant we use so-called circuit algebras, an extension of planar algebras which are the `right' setting to discuss virtual knots. Our tangle invariant is a circuit algebra morphism, and so behaves well under tangle operations and gives yet another definition for the Alexander polynomial. The MVA and the single variable Alexander polynomial are known to satisfy a number of relations, each of which has a proof relying on different approaches and techniques. Using our invariant we can give simple computational proofs of many of these relations, as well as an alternate proof that the MVA and our virtual extension are of finite type.

The focus of this dissertation is the interpolation and approximation of multivariate bandlimited functions via sampled (function) values. The first set of results investigates polynomial interpolation in connection with multivariate bandlimited functions. To this end, the concept of a uniformly invertible Riesz basis is developed (with examples), and is used to construct Lagrangian polynomial interpolants for particular classes of sampled square-summable data. These interpolants are used to derive two asymptotic recovery and approximation formulas. The first recovery formula is theoretically straightforward, with global convergence in the appropriate metrics; however, it becomes computationally complicated in the limit. This complexity is sidestepped in the second recovery formula, at the cost of requiring a more local form of convergence. The second set of results uses oversampling of data to establish a multivariate recovery formula. Under additional restrictions on the sampling sites and the frequency band, this formula demonstrates a certain stability with respect to sampling errors. Computational simplifications of this formula are also given.

This thesis investigates lattice compression of polynomial matrices over finite fields. For an m x n matrix, the goal of lattice compression is to find an m x (m+k) matrix, for some relatively small k, such that the lattice span of two matrices are equivalent. For any m x n polynomial matrix with degree bound d, it can be compressed by multiplying by a random n x (m+k) matrix B with degree bound s. In this thesis, we prove that there is a positive probability that L(A)=L(AB) with k(s+1)=\Theta(\log(md)). This is shown to hold even when s=0 (i.e., where B is a matrix of constants). We also design a competitive probabilistic lattice compression algorithm of the Las Vegas type that has a positive probability of success on any input and requires O~(nm^{\theta-1}B(d)) field operations.

Large integer factoring problem is a difficult computing problem. The security of many public-key cryptograohy system depend on the large interger factoring problem. Dr. Guan implement ¡uThe Multiple Polynomail Quadratic Sieve Algorithm¡v and name the program ¡uGQS¡v. The program successfully factor RSA-130 interger in 2004. It can reduce the time of sieving that the MPQS algorithm retain the smooth number with one or two prime. But finally the size of factor basis is large. We use some of the prime retained by the MPQS algorithm to match with the smooth number and reduce the size of factor basis. And then we can reduce the time of factoring. In this paper, we implement our idea in a AIX server and the result of this paper can be a suggestion of the improvement of MPQS.

Thesis (Ph. D.)--University of Oregon, 2001. / Typescript. Includes vita and abstract. Includes bibliographical references (leaf 49). Also available for download via the World Wide Web; free to University of Oregon users.

DC/DC buck converters are cascaded in order to generate proper load voltages. Rectified line voltage is normally converted to 48V, which then, by a bus voltage regulating converter also called the line conditioner converter, is converted to the bus voltage, e.g. 12V. A polynomial controller converter transforms the 12V into to a suitable load voltage, a fraction of or some few voltages. All cascaded converters are individually controlled in order to keep the output voltage stable constant. In this presentation focusing on the polynomial controller converter implemented as Ericsson’s buck converter BMR450. In this paper modeling, discretization and control of a simple Buck converter is presented. For the given DC-DC-Converter-Ericsson BMR 450 series, analyzing the disturbance properties of a second order buck converter controllers by a polynomial controller. The project is performed in Matlab and Simulink. The controller properties are evaluated for measurement noise, EMC noise and for parameter changes. / +46-762795822

We compute the E-polynomials of a family of twisted character varieties M [superscript g] (Sl [subscript n]) by proving they have polynomial count, and applying a result of N. Katz on the counting functions. To compute the number of F [subscript q]-points of these varieties as a function of q, we used a formula of Frobenius. Our calculations made use of the character tables of Gl [subscript n](q) and Sl subscript n](q), previously computed by J. A. Green and G. Lehrer, and a result of Hanlon on the Möbius function of a subposet of set-partitions. The Euler Characteristics of the M [superscript g] (Sl [subscript n]) are calculated then with these polynomial. / text