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SL₂-polynomial invariance

Description: Article on SL₂-polynomial invariance. The purpose of this article is to describe completely, in a constructive way, the structure of these invariant polynomials in the particular case n = 2, for an arbitrary field k.
Date: 1998
Creator: Anghel, Nicolae
Partner: UNT College of Arts and Sciences


Description: The GEORGE computer at Argonne National Laboratory was coded to transform yields as well as the coefficients between the two systems. The matrices used in the transformation of coefficients are tabulated. (auth)
Date: August 1, 1959
Creator: Lane, R.O.; Miller, W.F. & Hillstrom, K.E.
Partner: UNT Libraries Government Documents Department

Algebraic Number Fields

Description: This thesis investigates various theorems on polynomials over the rationals, algebraic numbers, algebraic integers, and quadratic fields. The material selected in this study is more of a number theoretical aspect than that of an algebraic structural aspect. Therefore, the topics of divisibility, unique factorization, prime numbers, and the roots of certain polynomials have been chosen for primary consideration.
Date: August 1991
Creator: Hartsell, Melanie Lynne
Partner: UNT Libraries

The Use of Chebyshev Polynomials in Numerical Analysis

Description: The purpose of this paper is to investigate the nature and practical uses of Chebyshev polynomials. Chapter I gives recognition to mathematicians responsible for studies in this area. Chapter II enumerates several mathematical situations in which the polynomials naturally arise and suggests reasons for the pursuance of their study. Chapter III includes: Chebyshev polynomials as related to "best" polynomial approximation, Chebyshev series, and methods of producing polynomial approximations to continuous functions. Chapter IV discusses the use of Chebyshev polynomials to solve certain differential equations and Chebyshev-Gauss quadrature.
Date: December 1975
Creator: Forisha, Donnie R.
Partner: UNT Libraries

A local construction of the Smith normal form of a matrix polynomial

Description: We present an algorithm for computing a Smith form with multipliers of a regular matrix polynomial over a field. This algorithm differs from previous ones in that it computes a local Smith form for each irreducible factor in the determinant separately and then combines them into a global Smith form, whereas other algorithms apply a sequence of unimodular operations to the original matrix row by row (or column by column). The performance of the algorithm in exact arithmetic is reported for several test cases.
Date: September 1, 2008
Creator: Wilkening, Jon & Yu, Jia
Partner: UNT Libraries Government Documents Department

Quadratic Forms

Description: This paper shall be mostly concerned with the development and the properties of three quadratic polynomials. The primary interest will by with n-ary quadratic polynomials, called forms.
Date: June 1959
Creator: Cadenhead, Clarence Tandy
Partner: UNT Libraries

On complexity of the mixed volume of parallelograms

Description: Let K = (K{sub 1}...K{sub n}) be a n-tuple of convex compact subsets in the Euclidean space R{sup n}, and let V({center_dot}) be the Euclidean volume in R{sup n}. It is well known Herman Minkowski result (see for instance 5), that the value of the V{sub K}({gamma}{sub 1}K{sub 1} + ... {gamma}{sub n}K{sub n}) is a homogeneous polynomial of degree n, called the Minkowski polynomial, in nonnegative variables {gamma}{sub 1}...{gamma}{sub n1}, where '+' denotes Minkowski sum, and {gamma}K denotes the dilatation of K with coefficient {gamma}. The coefficient V(K{sub 1}...K{sub n}) of {gamma}{sub 1}{center_dot}{gamma}{sub 2}...{center_dot}{gamma}{sub n} is called the mixed volume of K{sub 1}...K{sub n}. Alternatively, V(K{sub 1}...K{sub n}) = ({partial_derivative}{sup n} / {partial_derivative}{gamma}{sub 1}...{partial_derivative}{gamma}{sub n})V{sub K}({gamma}{sub 1}K{sub 1}+...{gamma}{sub n}K{sub n}).
Date: January 1, 2009
Creator: Gurvits, Leonid
Partner: UNT Libraries Government Documents Department

The Comparison and Evaluation of Three Fiber Composite Failure Criteria

Description: Three specific failure criteria for the transversely isotropic fiber composite case will be discussed. All three use the polynomial expansion method. The three criteria are the Tsai-Wu criterion, the Hashin criterion and the Christensen criterion. All three criteria will be given in forms that admit direct and easy comparison, which has not usually been done. The central differences between these three criteria will be discussed, and steps will be taken toward the evaluation of them.
Date: February 8, 2005
Creator: Christensen, R M
Partner: UNT Libraries Government Documents Department

Ádám's Conjecture and Its Generalizations

Description: This paper examines idam's conjuecture and some of its generalizations. In terms of Adam's conjecture, we prove Alspach and Parson's results f or Zpq and ZP2. More generally, we prove Babai's characterization of the CI-property, Palfy's characterization of CI-groups, and Brand's result for Zpr for polynomial isomorphism's. We also prove for the first time a characterization of the CI-property for 1 SG, and prove that Zn is a CI-Pn-group where Pn is the group of permutation polynomials on Z,, and n is square free.
Date: August 1990
Creator: Dobson, Edward T. (Edward Tauscher)
Partner: UNT Libraries

Canonicalization and Demodulation

Description: Mechanisms that were developed for the Argonne National Laboratory - Northern Illinois University theorem proving system are discussed. By defining special input clauses and demodulators, it is possible to simulate mathematical processes such as canonicalization of polynomials with no special programming. The mechanisms presented resulted from a study of the X³ = X problem in ring theory. The use of the mechanisms allowed this problem to the solved for the first time by the automated theorem proving system.
Date: February 1981
Creator: Veroff, Robert L.
Partner: UNT Libraries Government Documents Department

Polynomial Isomorphisms of Cayley Objects Over a Finite Field

Description: In this dissertation the Bays-Lambossy theorem is generalized to GF(pn). The Bays-Lambossy theorem states that if two Cayley objects each based on GF(p) are isomorphic then they are isomorphic by a multiplier map. We use this characterization to show that under certain conditions two isomorphic Cayley objects over GF(pn) must be isomorphic by a function on GF(pn) of a particular type.
Date: December 1989
Creator: Park, Hong Goo
Partner: UNT Libraries

Curved mesh generation and mesh refinement using Lagrangian solid mechanics

Description: We propose a method for generating well-shaped curved unstructured meshes using a nonlinear elasticity analogy. The geometry of the domain to be meshed is represented as an elastic solid. The undeformed geometry is the initial mesh of linear triangular or tetrahedral elements. The external loading results from prescribing a boundary displacement to be that of the curved geometry, and the final configuration is determined by solving for the equilibrium configuration. The deformations are represented using piecewise polynomials within each element of the original mesh. When the mesh is sufficiently fine to resolve the solid deformation, this method guarantees non-intersecting elements even for highly distorted or anisotropic initial meshes. We describe the method and the solution procedures, and we show a number of examples of two and three dimensional simplex meshes with curved boundaries. We also demonstrate how to use the technique for local refinement of non-curved meshes in the presence of curved boundaries.
Date: December 31, 2008
Creator: Persson, P.-O. & Peraire, J.
Partner: UNT Libraries Government Documents Department

Bessel-Zernike Discrete Variable Representation Basis

Description: The connection between the Bessel discrete variable basis expansion and a specific form of an orthogonal set of Jacobi polynomials is demonstrated. These so-called Zernike polynomials provide alternative series expansions of suitable functions over the unit interval. Expressing a Bessel function in a Zernike expansion provides a straightforward method of generating series identities. Furthermore, the Zernike polynomials may also be used to efficiently evaluate the Hankel transform for rapidly decaying functions or functions with finite support.
Date: October 24, 2005
Creator: Cerjan, C J
Partner: UNT Libraries Government Documents Department

Modified Gelfand-Tseltin patterns, lattice permutations, and skew-tableau polynomials

Description: A modification of the well-known Gelfand-Tsetlin patterns, which are one-to-one with Young-Weyl standard tableaux is introduced. These new patterns are in one-to-one correspondence with skew-tableaux, and with a slight modification can be used to enumerate lattice permutations. In particular the coupling rule for angular momentum takes an elementary form in terms of these modified patterns. These interrelations will be presented, together with an outline of the construction of a class of polynomials that generalizes the skew Schur functions.
Date: January 1, 2002
Creator: Louck, James D.
Partner: UNT Libraries Government Documents Department

Uncertainty quantification for large-scale ocean circulation predictions.

Description: Uncertainty quantificatio in climate models is challenged by the sparsity of the available climate data due to the high computational cost of the model runs. Another feature that prevents classical uncertainty analyses from being easily applicable is the bifurcative behavior in the climate data with respect to certain parameters. A typical example is the Meridional Overturning Circulation in the Atlantic Ocean. The maximum overturning stream function exhibits discontinuity across a curve in the space of two uncertain parameters, namely climate sensitivity and CO{sub 2} forcing. We develop a methodology that performs uncertainty quantificatio in the presence of limited data that have discontinuous character. Our approach is two-fold. First we detect the discontinuity location with a Bayesian inference, thus obtaining a probabilistic representation of the discontinuity curve location in presence of arbitrarily distributed input parameter values. Furthermore, we developed a spectral approach that relies on Polynomial Chaos (PC) expansions on each sides of the discontinuity curve leading to an averaged-PC representation of the forward model that allows efficient uncertainty quantification and propagation. The methodology is tested on synthetic examples of discontinuous data with adjustable sharpness and structure.
Date: September 1, 2010
Creator: Safta, Cosmin; Debusschere, Bert J.; Najm, Habib N. & Sargsyan, Khachik
Partner: UNT Libraries Government Documents Department

Predecessor and permutation existence problems for sequential dynamical systems

Description: A class of finite discrete dynamical systems, called Sequential Dynamical Systems (SDSs), was introduced in BMR99, BR991 as a formal model for analyzing simulation systems. An SDS S is a triple (G, F,n ),w here (i) G(V,E ) is an undirected graph with n nodes with each node having a state, (ii) F = (fi, fi, . . ., fn), with fi denoting a function associated with node ui E V and (iii) A is a permutation of (or total order on) the nodes in V, A configuration of an SDS is an n-vector ( b l, bz, . . ., bn), where bi is the value of the state of node vi. A single SDS transition from one configuration to another is obtained by updating the states of the nodes by evaluating the function associated with each of them in the order given by n. Here, we address the complexity of two basic problems and their generalizations for SDSs. Given an SDS S and a configuration C, the PREDECESSOR EXISTENCE (or PRE) problem is to determine whether there is a configuration C' such that S has a transition from C' to C. (If C has no predecessor, C is known as a garden of Eden configuration.) Our results provide separations between efficiently solvable and computationally intractable instances of the PRE problem. For example, we show that the PRE problem can be solved efficiently for SDSs with Boolean state values when the node functions are symmetric and the underlying graph is of bounded treewidth. In contrast, we show that allowing just one non-symmetric node function renders the problem NP-complete even when the underlying graph is a tree (which has a treewidth of 1). We also show that the PRE problem is efficiently solvable for SDSs whose state values are from ...
Date: January 1, 2002
Creator: Barrett, C. L. (Christopher L.); Hunt, H. B. (Harry B.); Marathe, M. V. (Madhav V.); Rosenkrantz, D. J. (Daniel J.) & Stearns, R. E. (Richard E.)
Partner: UNT Libraries Government Documents Department

Modal Analysis Using the Singular Value Decomposition

Description: Many methods exist for identifying modal parameters from experimental transfer function measurements. For frequency domain calculations, rational fraction polynomials have become the method of choice, although it generally requires the user to identify frequency bands of interest along with the number of modes in each band. This process can be tedious, especially for systems with a large number of modes, and it assumes the user can accurately assess the number of modes present in each band from frequency response plots of the transfer functions. When the modal density is high, better results can be obtained by using the singular value decomposition to help separate the modes before the modal identification process begins. In a typical calculation, the transfer function data for a single frequency is arranged in matrix form with each column representing a different drive point. The matrix is input to the singular value decomposition algorithm and left- and right-singular vectors and a diagonal singular value matrix are computed. The calculation is repeated at each analysis frequency and the resulting data is used to identify the modal parameters. In the optimal situation, the singular value decomposition will completely separate the modes from each other, so that a single transfer function is produced for each mode with no residual effects. A graphical method has been developed to simplify the process of identifying the modes, yielding a relatively simple method for computing mode shapes and resonance frequencies from experimental data.
Date: February 5, 2004
Creator: Fahnline, J. B.; Campbell, R. L. & Hambric, S. A.
Partner: UNT Libraries Government Documents Department

Polynomial Curve and Surface Fitting

Description: The main problems of numerical analysis involve performing analytical operations, such as integration, differentiation, finding zeroes, interpolation, and so forth, of a function when all the data available are some samples of the function. Therefore, the purpose of this paper is to investigate the following problem: given a set of data points (x[sub i], y[sub i]) which are samples of some function, determine an approximating function. Further, extend the problem to that of determining an approximating function for a surface given some samples (x[sub i], y[sub j], z[sub ij]) of the surface.
Date: January 1968
Creator: Capps, Ann Dowdy
Partner: UNT Libraries