| Multivariate Approximation |
29 DEC 1998 |
5 pages |
| Authors:
Carl DE Boor; Amos Ron; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
 | Methods for representing multi-dimensional objects, such as functionsof several variables and, more generally, (hyper-)surfaces is the main objective. One goal of such representation, whether approximate or exact, is theefficient evaluation of the object: Multivariate Polynomial Interpolation as well as Scattered Data Approximation both fall into this category. Another goal is a representation that allows one to identify and access easily and simultaneously relevant aspects of the object. The topics of ... |
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| Compactly Supported Tight Affine Spline Frames in L2(Rd) |
FEB 96 |
20 pages |
| Authors:
Amos Ron; Zuowei Shen; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| The theory of RS2 is applied to yield compactly supported tight affine frames (wavelets) in L2(Rd) from box splines. The wavelets obtained are smooth piecewise polynomials on a simple mesh; furthermore, they exhibit a wealth of symmetries, and have a relatively small support. The number of 'mother wavelets', however, increases with the increase of the required smoothness. Two bivariate constructions, of potential practical value, are highlighted. In both, the wavelets ... |
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| Stability and Independence of the Shifts of Finitely Many Refinable Functions |
FEB 96 |
24 pages |
| Authors:
Thomas A. Hogan; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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| Typical constructions of wavelets depend on the stability of the shifts of an underlying refinable function. Unfortunately, several desirable properties are not available with compactly supported orthogonal wavelets, e.g. symmetry and piecewise polynomial structure. Presently, multiwavelets seem to offer a satisfactory alternative. The study of multiwavelets involves the consideration of the properties of several (simultaneously) refinable functions. In Section 2 of this paper, we characterize stability and linear independence of ... |
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| Affine Systems in L2(Rd): The Analysis of the Analysis Operator |
DEC 95 |
35 pages |
| Authors:
Amos Ron; Zuowei Shen; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| Discrete affine systems are obtained by applying dilations to a given shift-invariant system. The complicated structure of the affine system is due, first and foremost, to the fact that it is not invariant under shifts. Affine frames carry the additional difficulty that they are 'global' in nature: it is the entire interaction between the various dilation levels that determines whether the system is a frame, and not the behaviour of ... |
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| Gramian Analysis of Affine Bases and Affine Frames |
APR 95 |
10 pages |
| Authors:
Amos Ron; Zuowei Shen; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| Shift invariance fiberization techniques are applied for the study of the synthesis and analysis operators of affine (wavelet) systems. In this approach, one has first to circumvent the fact that affine systems are not shift invariant. The results obtained include characterizations of the Bessel property, the Riesz basis property and the frame property of such sets in terms of the behaviour of simpler operators. Various estimates of the lower and ... |
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| Wave Interactions and Variation Estimates For Self-Similar Viscous Limits in Systems of Conservation Laws |
FEB 95 |
66 pages |
| Authors:
Athanasios E. Tzavaras; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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| We consider the problem of self similar viscous limits for general systems of conservation laws. First, we give conditions so that the resulting boundary value problem admits solutions. In particular this covers the class of symmetric hyperbolic systems. Second, we show that if the system is strictly hyperbolic and the Riemann data are sufficiently close then the resulting family of solutions is of uniformly bounded variation and oscillation. Third, we ... |
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| Shock Profiles and Self-Similar Fluid Dynamic Limits |
DEC 94 |
14 pages |
| Authors:
Marshall Slemrod; Athanasios E. Tzavaras; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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| Consider the fluid dynamic limit problem for the Broadwell system of the kinetic theory of gases, for Riemann, Maxwellian initial data. The formal limit is the Riemann problem for a pair of conservation laws and is invariant under dilations of coordinates. We are interested on the structure of shock solutions entailed by fluid-dynamic limits. We review certain results on the existence of shock profiles for the Broadwell model and on ... |
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| Weyl-Heisenberg Frames and Riesz Bases in L2(Rd) |
OCT 94 |
42 pages |
| Authors:
Amos Ron; Zuowei Shen; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| We study Weyl-Heisenberg (= Gabor) expansions for either L2(Rd) or a subspace of it. These are expansions in terms of the spanning set, involving K and L are some discrete lattices in Rd, P, in L2(Rd), is finite, E is the translation operator, and M is a modulation operator. Such sets X are known as WH systems. The analysis of the 'basis' properties of WH systems (e.g. being a frame ... |
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| Numerical Simulations of Cluster Formation Using a Discrete Velocity Kinetic Theory of Gases |
07 APR 94 |
43 pages |
| Authors:
M. Slemrod; A. Qi; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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| Cluster formation is simulated numerically with discrete velocity Boltzmann model in two space dimensions. The model exhibits cluster coagulation, fragmentation, and transport. It evolves on two different scales obtained from an elastic and inelastic collision Knudsen numbers respectively. For flow impinging on a wall with specularly reflective boundary condition these scales appear both analytically and numerically. |
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| Gauss Elimination by Segments and Multivariate Polynomial Interpolation |
APR 94 |
24 pages |
| Authors:
C. DE Boor; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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| The construction of a polynomial interpolant to data given at finite pointsets (or, most generally, to data specified by finitely many linear functionals) is considered, with special emphasis on the linear system to be solved. Gauss elimination by segments(i.e., by groups of columns rather than by columns) is proposed as a reasonable means for obtaining a description of all solutions and for seeking out solutions with 'good' properties. A particular ... |
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| Elastic as Limit of Viscoelastic Response, in a Context of Self-Similar Viscous Limits |
MAR 94 |
38 pages |
| Authors:
Anthanasios E. Tzavaras; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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| We study the equations of one-dimensional isothermal elastic response as the small viscosity limit of the equations of viscoelasticity, in a context of self-similar viscous limits for Riemann data. THe limiting procedure is justified and a solution of the Riemann problem for the equations of elasticity is obtained. The emerging solution is composed of two wave fans, each consisting of rarefactions, shocks and contact discontinuities, separated by constant states. At ... |
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| Frames and Stable Bases for Shift-Invariant Subspaces of L2(IRd) |
FEB 94 |
44 pages |
| Authors:
Amos Ron; Zuowei Shen; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| We study in this paper certain types of bases for shift-invariant subspaces. Our primary objective is to connect among three important families of basis sets: shift-invariant sets. Weyl-Heisenberg sets, and affine (wavelet) sets. The present paper is the first in a series of three, and is concerned with the basic theory of shift-invariant bases for the shift-invariant spaces. The two papers, (RS1) and (RS2), will focus on the applications of ... |
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| Strictly Positive Definite Functions on Spheres |
FEB 94 |
23 pages |
| Authors:
Amos Ron; Xingping Sun; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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| In this paper we study strictly positive definite functions on the unit sphere of the m-dimensional Euclidean space. Such functions can be used for solving a scattered data interpolation problem on spheres. Since positive definite functions on the sphere were already characterized by Schoenberg some fifty years ago, the issue here is to determine what kind of positive definite functions are actually strictly positive definite. The study of this problem ... |
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| Oscillations in Piston-Driven Shear Flow of a Non-Newtonian Fluid |
FEB 94 |
17 pages |
| Authors:
David S. Malkus; John A. Nohel; Bradley J. Plohr; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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| In recent experiments on piston-driven shear flow of a highly elastic and very viscous non-Newtonian fluid. Lim and Schowalter observed nearly periodic oscillations in the particle velocity at the channel wall for particular values of the constant volumetric flow rate. Such periodicity has been characterized as a 'stick-/slip' phenomenon caused by the failure of the fluid to adhere to the wall. We suggest an alternative explanation for these oscillations using ... |
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| Radial Base Functions: Lp-approximation Orders with Scattered Centres |
JAN 94 |
22 pages |
| Authors:
Martin D. Buhmann; Amos Ron; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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| In this paper we generalize several results on uniform approximation orders with radial basis functions in (Buhmann, Dyn and Levin, 1993) and (Dyn and Ron, 1993) to Lp-approximation orders. These results apply, in particular, to approximants from spaces spanned by translates of radial basis functions by scattered centres. Examples to which our results apply include quasi- interpolation and least-squares approximation from radial function spaces. Approximation order, Scattered translates, Radial basis ... |
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| Multivariate Spline Approximation |
22 DEC 93 |
6 pages |
| Authors:
Carl R. De Boor; Amos Ron; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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| We started out with the goal of understanding approximation order in a multivariate context, including the approximation of surfaces. In addition, we wanted to understand better the use and analysis of our approach to multivariate polynomial interpolation. We ended up concentrating on approximation from shift- invariant spaces of functions on IR real space. Here, S is shift-invariant if f is an element S implies that also f(. - alpha) is ... |
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| Radial Basis Function Approximation: From Gridded Centers to Scattered Centers |
DEC 93 |
32 pages |
| Authors:
Nira Dyn; Amos Ron; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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| The paper studies L-infinity (IR d)-norm approximations from a space spanned by a discrete set of translates of a basis function theta. Attention here is restricted to functions theta whose Fourier transform is smooth on IRd/ 0, and has a singularity at the origin. Examples of such basis functions are the thin-plate splines and the multiquadrics, as well as other types of radial basis functions that are employed in Approximation ... |
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| Progress Report for ONR Grant N00014-93-1-0015 (University of Wisconsin) |
15 JUL 93 |
4 pages |
| Authors:
WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| The research effort of Slemrod has been directed to developing a discrete velocity kinetic theory model for liquid-vapor phase transitions. The aim of the research is to develop a 'basic principles' description of liquid- vapor phase transitions (such as those occurring in an internal combustion engine) which is not a priori biased by ad hoc equations of state (constitutive relations). The modelling is almost completely finished, numerical simulations are being ... |
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| Wave Structure Induced by Fluid Dynamic Limits in the Broadwell Model |
JUL 93 |
44 pages |
| Authors:
Athanasios E. Tzavaras; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| Consider the fluid dynamic limit problem for the Broadwell system of the kinetic theory of gases, for Riemann, Maxwellian initial data. The formal limit is the Riemann problem for a pair of conservation laws and is invariant under dilations of coordinates. The approach of self-similar fluid dynamic limits consists in replacing the mean free path in the Broadwell model so that the resulting problem preserves the invariance under dilations. The ... |
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| Approximation Orders of and Approximation Maps from Local Principal Shift-Invariant Spaces |
MAY 93 |
23 pages |
| Authors:
Amos Ron; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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| Approximation orders of shift-invariant subspaces generated by the shifts of one compactly supported function are considered. In that course, explicit approximation maps are constructed. The approach avoids quasi- interpolation and applies to stationary and non-stationary refinements. The general results are specialized to box spline spaces, to obtain new results on their approximation orders. |
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| Self-Similar Fluid Dynamic Limits for the Broadwell System |
OCT 92 |
60 pages |
| Authors:
Marshall Slemrod; Athanasios E. Tzavaras; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| This report discusses a new approach for the resolution of the fluid dynamic limit for the Broadwell system of the kinetic theory of gases, appropriate in the case of Riemann, Maxwellian data. Since the formal limiting system is expected to have self-similar solutions, we are motivated to replace the Knudsen number E in the Broadwell model so that the resulting model admits self-similar solutions in E = x/t and then ... |
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| Multiresolution Analysis by Infinitely Differentiable Compactly Supported Functions |
SEP 92 |
12 pages |
| Authors:
N. Dyn; A. Ron; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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| The paper is concerned with the introduction and study of multiresolution analysis based on the up function, which is an infinitely differentiable function supported on (0,2). Such analysis is, necessarily, nonstationary. It is shown that the approximation orders associated with the corresponding spaces are spectral, thus making the spaces attractive for the approximation of very smooth functions. |
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| Characterizations of Linear Independence and Stability of the Shifts of a Univariate Refinable Function in Terms of Its Refinement Mask |
SEP 92 |
18 pages |
| Authors:
Amos Ron; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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| Characterizations of the linear independence and stability properties of the integer translates of a compactly supported univariate refinable function in terms of its mask are established. The results extend analogous ones of Jia and Wang which were derived for dyadic refinements and finite masks. |
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| Hyperbolic Conservation Laws with Umbilic Points I |
AUG 92 |
61 pages |
| Authors:
Gui-Qiang Chen; Pui T. Kan; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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| In this paper a compactness framework for approximate solutions to nonlin- ear hyperbolic systems with umbilic points is established by combining ideas in modern nonlinear analysis with classical methods, and by a detailed analysis of a highly singular Euler-Poisson-Darboux-type equation. Then this framework is successfully applied to prove the convergence of the Lax-Friedrichs scheme, the Godunov scheme and the viscosity method, and the existence of global entropy solutions for the ... |
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| On Ascertaining Inductively the Dimension of the Joint Kernel of Certain Commuting Linear Operators |
JUN 92 |
32 pages |
| Authors:
Carl DE Boor; Amos Ron; Zuowei Shen; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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| Given a linear space S (over some field), we attempt to determine the dimension of spaces of a certain form with a L a (finite) sequence of linear endomorphisms of S, i.e., a sequence in L(S). |
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| Nonlinear Analysis Techniques for Shear Band Formations at High Strain- Rates |
APR 92 |
26 pages |
| Authors:
Athanasios E. Tzavaras; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| One of the most striking manifestations of instability in solid mechanics is the localization of shear strain into narrow bands during high speed, plastic deformations of metals. According to one theory, the formation of shear bands is attributed to effective strain-softening response, which results at high strain rates as the net outcome of the influence of thermal softening on the, normally, strain-hardening response of metals. Our objective is to review ... |
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| On the Construction of Multivariate (pre) Wavelets |
FEB 92 |
43 pages |
| Authors:
Carl DE Boor; Ronald A. DeVore; Amos Ron; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| A new approach for the construction of wavelets and prewavelets on IR d from multiresolution is presented. The method uses only properties of shift- invariant spaces and orthogonal projectors from L2(IRd) onto these spaces, and requires neither decay nor stability of the scaling function. Furthermore, this approach allows a simple derivation of previous, as well as new, constructions of wavelets, and leads to a complete resolution of questions concerning the ... |
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| The Structure of Finitely Generated Shift-Invariant Spaces in L2(IR(d)) |
FEB 92 |
35 pages |
| Authors:
Carl DE Boor; Ronald A. DeVore; Amos Ron; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| A simple characterization is given of finitely generated subspaces of L2(IR(d)) which are invariant under translation by any (multi)integer, and used to give conditions under which such a space has a particularly nice generating set, namely a basis, and, more than that, a basis with desirable properties, such as stability, orthogonality, or linear independence. The last property makes sense only for 'local' spaces, i.e., shift-invariant spaces generated by finitely many ... |
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| Approximation from Shift-Invariant Subspaces of L sup 2 (R sup d) |
06 JUL 91 |
22 pages |
| Authors:
Carl DE Boor; Ronald A. DeVore; Amos Ron; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| A complete characterization is given of closed shift-invariant subspaces of which provide a specified approximation order. When such a space is principal (i.e., generated by a single function), then this characterization is in terms of the Fourier transform of the generator. As a special case, we obtain the classical Strang-Fix conditions, but without requiring the generating function to decay at infinity. The approximation order of a general closed shift-invariant space ... |
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| Fourier Analysis of the Approximation Power of Principal Shift-Invariant Spaces |
JUL 91 |
36 pages |
| Authors:
Carl DE Boor; Amos Ron; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| Spaces spanned by finitely or countably many translates of one or several basic functions play an important role in spline theory, radial basis function theory, sampling theory and wavelet theory. Spline theory stresses the case when the basic functions are compactly supported, while sampling theory single out the case when the spectrum (i.e., the support of the Fourier transform) of the basic functions is compact. In the radial basis function ... |
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| Some Problems in Nonlinear Analysis |
30 MAY 91 |
4 pages |
| Authors:
M. G. Crandall; P. H. Rabinowitz; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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| Problems in Nonlinear Continuum Dynamics |
14 MAY 91 |
7 pages |
| Authors:
Marshall Slemrod; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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| Slemrod's research in 1990-1991 centered on two issue: (1) the kinetics of coagulation processes, (2) behavior of discrete velocity models in the kinetic theory of gases. In the first area Slemrod has (a) given a new method for solving the special class of coagulation equations which exhibit gelatin and (b) derived and proved existences of similarity solutions for coagulation equations with diffusion. In the second area Slemrod has used his ... |
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| Principal Components of Minus M-Matrices |
FEB 91 |
24 pages |
| Authors:
Michael Neumann; Hans Schneider; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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| This paper determines the nonnegativity of the principal components of an n x n nonnegative matrix P in terms of the marked reduced graph R(A) of A = P - rho(P)I, the minus M matrix which can be associated with P. We then apply this result to consider various types of nonnegative bases for the Perron eigenspace of P which can be extracted from a certain nonnegative matrix which is ... |
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| Interdisciplinary Research on Viscoelasticity and Rheology |
14 DEC 90 |
11 pages |
| Authors:
John A. Nohel; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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| Viscoelastic materials with fading memory, e.g. polymers, suspensions, emulsions, exhibit behavior that is intermediate between the nonlinear hyperbolic response of purely elastic materials and the strongly diffusive, parabolic response of viscous fluids. The following problems that are an outgrowth of earlier research were investigated during the reporting period: Provide numerical and analytic explanation of several striking phenomena observed in experiments in shear flows of highly elastic and very viscous non- ... |
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| Analytical Solution of Plane Poiseuille Flow of a Johnson-Segalman Fluid |
MAY 90 |
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| Authors:
M. Yao; D. S. Malkus; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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 | In this paper the pressure driven plane Poiseuille flow of th Johnson-Segalman fluid is studied. By changing integral variable and solving a cubic equation at each location, we are able to obtain the exact steady solutions for this flow. Both monotone and non-monotone stress-strain-rate relations are considered and complete formulation & solution procedures are developed. The flow curves from Vinogradov et al's spurt data capillary flow are simulated analytically with ... |
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| Numerical Simulation of Hole Pressure for a Johnson-Segalman Fluid |
APR 90 |
27 pages |
| Authors:
M. Yao; D. S. Malkus; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| In this paper we study the hole pressure problem for plane, steady, creeping shear flows of a Johnson Segalman model. To correctly apply the theory of Higashitani, Pritchard, Baird & Lodge (HPBL), we start with a modified hole pressure relation (MHPR) and we simulate the hole pressure measurement by FEM and multi mesh extrapolation techniques. The path integrals of MHPR & HPBL are evaluated and a full instrument simulation is ... |
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| Strain Softening in Viscoelasticity of the Rate Type |
MAR 90 |
34 pages |
| Authors:
Athanasios E. Tzavaras; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| A motivation for studying this problem stems from a program of understanding the phenomenon of shear band formation at high strain rates. Shear bands are narrow regions of intensely concentrated shearing deformation that are observed during the plastic deformation of many materials. The occurrence of shear bands is typically associated with strain softening type response, past a critical strain, of the measured average shear stress versus the measured average shear ... |
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| A Variational Approach to Heteroclinic Orbits for a Class of Hamiltonian Systems |
FEB 90 |
17 pages |
| Authors:
Paul H. Rabinowitz; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| A large literature has developed in the last decade in which methods from the calculus of variations have been used to prove the periodic solutions of Hamiltonian systems of ordinary differential equations. The recent monograph of Mawhin and Willem provides a sizable bibliography of such works. Aside from equilibria, periodic solutions are the simplest global in time solutions of differential equations. It is only within the past one - two ... |
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| Spline Functions and Surfaces |
90 |
8 pages |
| Authors:
Carl DE Boor; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| The goal was to study the use of (smooth) piecewise polynomial spaces for the approximation of functions in one and, preferably, in several variables and find a better understanding of how well one can approximate from specific spaces, for specific schemes for approximation, including good bases for such spaces, and to make inroads on the problem of extending techniques for curve fitting by smoothly patched curves to surface interpolation. Progress ... |
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| Error Cancellation in HPBL Derivation of Elastic Hole-Pressure Error |
DEC 89 |
23 pages |
| Authors:
M. Yao; D. S. Malkus; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| Analytical studies of the hole-pressure error for non-Newtonian creeping flows over a transverse slot are pursued with particular interest in the theory of Higashitani, Pritchard, Baird and Lodge (HPBL). To correctly apply the HPBL theory a modified hole-pressure relation (MHPR) is employed. Some important mathematical properties of the MHPR are presented. By studying the MHPR in streamline coordinate formulation we find a fortuitous error cancellation phenomenon in the derivation of ... |
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| Some Results on Connecting Orbits for a Class of Hamiltonian Systems |
17 OCT 89 |
34 pages |
| Authors:
Paul H. Rabinowitz; Kazunaka Tanaka; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| The existence of various kinds of connecting orbits is established for a certain Hamiltonian system as well as its time dependent analogue. For the autonomous case, our main assumption is that V has a global maximum, e.g. at X = O and we find a various kinds of orbits terminating at O. For the time dependent case V has a local but not global maximum at X = O and ... |
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| New Transient Algorithms for Non-Newtonian Flows |
11 OCT 89 |
31 pages |
| Authors:
David S. Malkus; Yi-Cheng Tsai; Robert W. Kolkka; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| A fully dynamic method for shear flows is presented that treats the short time-scales associated with Newtonian viscosity (or short relaxation processes) and shear-wave propagation implicitly, while treating the long relaxation processes explicitly. The method is generalized to flows with non- constant strain-rate histories in the context of the well-known fiber-drawing problem. The linearized stability of the methods is analyzed, and extension of these methods to planar flows is given. ... |
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| Quasi Interpolants and Approximation Power of Multivariate Splines |
25 SEP 89 |
36 pages |
| Authors:
de Boor Carl; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| The determination of the approximation power of spaces of multivariate splines with the aid of quasi interpolants is reviewed. In the process, streamlined description of the existing quasi interpolants theory is given. The author begin with a brief review of the approximation power of univariate splines since the techniques for its investigation are also those with which people have tried to understand the multivariate setup. (That may in fact be ... |
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| Interdisciplinary Research in Viscoelasticity and Rheology |
25 AUG 89 |
13 pages |
| Authors:
David S. Malkus; John A. Nohel; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| A deep understanding of viscoelasticity and rheology is crucial to advanced materials engineering and process design. Examples of such advanced materials are high-strength polymers and additives for lubricants; process design problems include spinning of synthetic fibers and injection molding. The materials involved in these technologies are often highly elastic and very viscous. As a consequence, they often display behavior intermediate between that of a solid and that of a fluid, ... |
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| Periodic Solutions of Hamiltonian Systems of 3-Body Type |
AUG 89 |
107 pages |
| Authors:
Abbas Bahri; Paul H. Rabinowitz; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| The study of time periodic solutions of the n-body problem is a classical one. See e.g. (1). The authors' goal in this paper is to present some new variational approaches of a global nature to a class of problems of 3-body type. Keywords: Calculus of variations; Theorems; Computations. (KR) |
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| Quadratic Dynamical Systems Describing Shear Flow of Non-Newtonian Fluids |
AUG 89 |
19 pages |
| Authors:
D. S. Malkus; J. A. Nohel; B. J. Plohr; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| Phase-plane techniques are used to analyze a quadratic system of ordinary differential equations that approximates a single relaxation-time system of partial differential equations used to model transient behavior of highly elastic non-Newtonian liquids in shear flow through slit dies. The latter one-dimensional model is derived from three-dimensional balance laws coupled with differential constitutive relations well-known by rheologists. The resulting initial-boundary-value problem is globally well-posed and possesses the key feature: the ... |
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| Analysis of New Phenomena in Shear Flow of Non-Newtonian Fluids |
AUG 89 |
38 pages |
| Authors:
David S. Malkus; John A. Nohel; Bradley J. Plohr; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| Phase-plane and small-parameter asymptotic techniques are used to analyze systems of ordinary differential equations that describe the transient behavior of non-Newtonian fluids in shear flow. These systems approximate the partial differential equations that derive from three-dimensional balance laws and from differential constitutive models for highly elastic liquids. Two models are considered: one with a single relaxation time and small Newtonian viscosity; the other with two relaxation times and no Newtonian ... |
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| Stability of Discontinuous Shearing Motions of a Non-Newtonian Fluid |
18 JUL 89 |
11 pages |
| Authors:
J. A. Nohel; R. L. Pego; A. E. Tzavaras; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| This paper discusses recent results on the nonlinear stability of discontinuous steady states of a model initial-boundary value problem in one space dimension for incompressible, isothermal shear flow of a non-Newtonian fluid between parallel plates located at x = + or - 1, and driven by a constant pressure gradient. The non-Newtonian contribution to the shear stress is assumed to satisfy a simple differential constitutive law. The key feature is ... |
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| Stability of Discontinuous Steady States in Shearing Motion of a Non- Newtonian Fluid |
18 JUL 89 |
24 pages |
| Authors:
John A. Nohel; Robert L. Pego; Athanasios E. Tzavaras; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
|
| This document studies the nonlinear stability of discontinuous steady states of a model initial-boundary value problem in one space dimension for incompressible, isothermal shear flow of a non-Newtonian fluid driven by a constant pressure gradient. The non-Newtonian contribution to the shear stress is assumed to satisfy a simple differential constitutive law. The key feature is a non-monotone relation between the total steady shear stress and shear strain- rate that results ... |
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| Periodic Solutions of Spatially Periodic Hamiltonian Systems |
10 JUL 89 |
26 pages |
| Authors:
Patricio L. Felmer; WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES
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| This work is concerned with the study of existence and multiplicity of periodic solutions of Hamiltonian systems of ordinary differential equations z=J(Hz(z,t) + f(t)) when the Hamiltonian H(z,t) = H(p,q,t) is periodic in the variable q and superlinear in the variable p. By imposing a growth condition on the derivative of H, we obtain the existence of at least n + 1 periodic solutions, where n is the dimension of ... |
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