WISCONSIN UNIV MADISON MATHEMATICS RESEARCH CENTER
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Works:  2,683 works in 2,786 publications in 1 language and 3,048 library holdings 

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Most widely held works by
WISCONSIN UNIV MADISON MATHEMATICS RESEARCH CENTER
Periodic solutions of Hamiltonian systems : a survey by
Paul H Rabinowitz(
Book
)
4 editions published between 1977 and 1984 in English and held by 5 WorldCat member libraries worldwide
The existence of periodic solutions of Hamiltonian systems of ordinary differential equations is proved in various settings. A case in which energy is prescribed is treated in Section 1. Both free and forced vibration problems, where the period is fixed, are studied in Section 2. The proofs involve finite dimensional approximation arguments, variational methods, and appropriate estimates. (Author)
4 editions published between 1977 and 1984 in English and held by 5 WorldCat member libraries worldwide
The existence of periodic solutions of Hamiltonian systems of ordinary differential equations is proved in various settings. A case in which energy is prescribed is treated in Section 1. Both free and forced vibration problems, where the period is fixed, are studied in Section 2. The proofs involve finite dimensional approximation arguments, variational methods, and appropriate estimates. (Author)
Linear transformations of a functional integral, ii(
Book
)
2 editions published in 1962 in English and held by 4 WorldCat member libraries worldwide
It was proved in Seidman, T.I., Linear Transformations of a Functional Integral, I; Comm. Pure and Appl. Math., Vol. XII, No. 4 (1959), that the measure on the countable direct product of real lines wth identical normally distributed measures, transforms (with a specified RadonNikodym derivative) to an equivalent (mutually absolutely continuous) measure under li ear transformations of the form T = I + A with A a nonsingular, HilbertSchmidt operator with finite trace (evaluated with respect to the canonical basis). We shall extend this result to transformations of the form T = U(I + A) where U is unitary and A nonsingular and HilbertSchmidt but with no traceability condition imposed. (Author)
2 editions published in 1962 in English and held by 4 WorldCat member libraries worldwide
It was proved in Seidman, T.I., Linear Transformations of a Functional Integral, I; Comm. Pure and Appl. Math., Vol. XII, No. 4 (1959), that the measure on the countable direct product of real lines wth identical normally distributed measures, transforms (with a specified RadonNikodym derivative) to an equivalent (mutually absolutely continuous) measure under li ear transformations of the form T = I + A with A a nonsingular, HilbertSchmidt operator with finite trace (evaluated with respect to the canonical basis). We shall extend this result to transformations of the form T = U(I + A) where U is unitary and A nonsingular and HilbertSchmidt but with no traceability condition imposed. (Author)
Approximation by Smooth Bivariate Splines on a ThreeDirection Mesh by
R.Q Jia(
Book
)
3 editions published between 1983 and 1984 in English and held by 4 WorldCat member libraries worldwide
Univariate splines have been proved quite useful in practice. However, if one wants to fit a surface, or solve a partial differential equation numerically, one would naturally think of using multivariate splines. Here splines still mean piecewise polynomial functions. In this respect, a basic question is to ascertain, for a given mesh delta and a family S of splines on delta, what its optimal approximation order is. This question is challenging even for a regular triangular mesh delta, as soon as one demands that the approximating functions have a certain amount of smoothness. The report records a step toward answering the above question. (Author)
3 editions published between 1983 and 1984 in English and held by 4 WorldCat member libraries worldwide
Univariate splines have been proved quite useful in practice. However, if one wants to fit a surface, or solve a partial differential equation numerically, one would naturally think of using multivariate splines. Here splines still mean piecewise polynomial functions. In this respect, a basic question is to ascertain, for a given mesh delta and a family S of splines on delta, what its optimal approximation order is. This question is challenging even for a regular triangular mesh delta, as soon as one demands that the approximating functions have a certain amount of smoothness. The report records a step toward answering the above question. (Author)
Estimation of Variance of the Ratio Estimator by
Mathematics Research Center (United States. Army)(
Book
)
3 editions published between 1981 and 1984 in English and held by 4 WorldCat member libraries worldwide
A general class of estimators of the variance of the ratio estiamtor is considered, which includes two standard estimators V sub 0 and V sub 2 and approximates another estiamtor V sub H. Asymptotic expansions for the variances and biases of the proposed estimators are obtained. Based on this optimal variance estimator in the class is obtained and compared to the relative merits of three estimators V sub 0, V sub 1 and V sub 2 without any model assumption. Under a simple regression model a more definite comparison of V sub 0, V sub 1 and V sub 2 is made in terms of variance and bias
3 editions published between 1981 and 1984 in English and held by 4 WorldCat member libraries worldwide
A general class of estimators of the variance of the ratio estiamtor is considered, which includes two standard estimators V sub 0 and V sub 2 and approximates another estiamtor V sub H. Asymptotic expansions for the variances and biases of the proposed estimators are obtained. Based on this optimal variance estimator in the class is obtained and compared to the relative merits of three estimators V sub 0, V sub 1 and V sub 2 without any model assumption. Under a simple regression model a more definite comparison of V sub 0, V sub 1 and V sub 2 is made in terms of variance and bias
The general paraboloidal coordinate system(
Book
)
2 editions published in 1962 in English and held by 4 WorldCat member libraries worldwide
2 editions published in 1962 in English and held by 4 WorldCat member libraries worldwide
Efficient timestepping methods for miscible displacement problems with nonlinear boundary conditions by
Richard E Ewing(
Book
)
3 editions published in 1979 in English and held by 3 WorldCat member libraries worldwide
Efficient multistep procedures for timestepping Galerkin methods for nonlinear parabolic partial differential equations with nonlinear Neumann boundary conditions are presented and analyzed. The procedures involve using a preconditioned iterative method for approximately solving the different linear equations arising at each time step in a discrete time Galerkin method. Optimal order convergence rates are obtained for the iterative methods. Work estimates of almost optimal order are obtained. Many mathematical models for heat flow or fluid flow involve the specification of a flow rate across the boundary of a region which may depend in a nonlinear fashion upon the unknown variable (e.g. temperature). Formulation and analysis of efficient numerical procedures for approximating the solutions of such problems are studied. Previously, finite element methods used for modeling these physical problems were at most second order correct in the timediscretization error. Methods are produced which are second, third, and fourth order correct in time and which convert the nonlinear problems into solution of large systems of linear equations via an extremely stable algorithm with essentially no restrictions between sizes of time and space discretizations. The paper also contains work estimates which show the large computational savings of the preconditioned iterative stabilization technique. Almost optimal order work estimates are obtained
3 editions published in 1979 in English and held by 3 WorldCat member libraries worldwide
Efficient multistep procedures for timestepping Galerkin methods for nonlinear parabolic partial differential equations with nonlinear Neumann boundary conditions are presented and analyzed. The procedures involve using a preconditioned iterative method for approximately solving the different linear equations arising at each time step in a discrete time Galerkin method. Optimal order convergence rates are obtained for the iterative methods. Work estimates of almost optimal order are obtained. Many mathematical models for heat flow or fluid flow involve the specification of a flow rate across the boundary of a region which may depend in a nonlinear fashion upon the unknown variable (e.g. temperature). Formulation and analysis of efficient numerical procedures for approximating the solutions of such problems are studied. Previously, finite element methods used for modeling these physical problems were at most second order correct in the timediscretization error. Methods are produced which are second, third, and fourth order correct in time and which convert the nonlinear problems into solution of large systems of linear equations via an extremely stable algorithm with essentially no restrictions between sizes of time and space discretizations. The paper also contains work estimates which show the large computational savings of the preconditioned iterative stabilization technique. Almost optimal order work estimates are obtained
Simple Computable Bounds for Solutions of Linear Complementarity Problems and Linear Programs by
Olvi L Mangasarian(
Book
)
2 editions published in 1983 in English and held by 3 WorldCat member libraries worldwide
Surprisingly simple bounds are given for solutions of fundamental constrained optimization problems such as linear and convex quadratic programs. It is shown that every nonoptimal primaldual feasible point carries within it simple numerical information which bounds some or all components of all solution vectors. The results given permit one to compute bounds without even solving the optimization problems. (Author)
2 editions published in 1983 in English and held by 3 WorldCat member libraries worldwide
Surprisingly simple bounds are given for solutions of fundamental constrained optimization problems such as linear and convex quadratic programs. It is shown that every nonoptimal primaldual feasible point carries within it simple numerical information which bounds some or all components of all solution vectors. The results given permit one to compute bounds without even solving the optimization problems. (Author)
On lancaster's decomposition of a differential matricial operator(
Book
)
1 edition published in 1961 in English and held by 3 WorldCat member libraries worldwide
1 edition published in 1961 in English and held by 3 WorldCat member libraries worldwide
On A Neutral Functional Differential Equation in a Fading Memory Space by
Olof J Staffans(
Book
)
2 editions published between 1981 and 1983 in English and held by 3 WorldCat member libraries worldwide
The linear autonomous, neutral system of functional differential equations are studied in a fading memory space. Conditions are given which imply that solutions of the functional differential equation can be decomposed into a stable part and an unstable part. These conditions are of frequency domain type. The results can be used to decompose the semigroup generated by the functional differential equation into a stable part and an unstable part
2 editions published between 1981 and 1983 in English and held by 3 WorldCat member libraries worldwide
The linear autonomous, neutral system of functional differential equations are studied in a fading memory space. Conditions are given which imply that solutions of the functional differential equation can be decomposed into a stable part and an unstable part. These conditions are of frequency domain type. The results can be used to decompose the semigroup generated by the functional differential equation into a stable part and an unstable part
A FreeBoundary Problem for a Degenerate Parabolic System by
Emmanuele DiBenedetto(
Book
)
2 editions published between 1981 and 1983 in English and held by 3 WorldCat member libraries worldwide
The degenerate parabolic system (1.1) in the introduction, serves as a model for heat conduction in a heterogeneous medium consisting of two components. The first component is made up of small pieces suspended in the second component, and the second component undergoes a change of phase at a prescribed temperature. This phenomenon occurs in a mixture of gravel and wet soil (for example, melting of frozen soil). Existence and uniqueness results of weak solutions of the degenerate parabolic problem are shown by employing monotone operator theory. Local regularity, such as continuity and boundedness of the solution is studied. A discussion is provided about the mutual interplay of the thermodynamic temperature (the temperature in the first component) and the conductive temperature (the temperature in the second component. (Author)
2 editions published between 1981 and 1983 in English and held by 3 WorldCat member libraries worldwide
The degenerate parabolic system (1.1) in the introduction, serves as a model for heat conduction in a heterogeneous medium consisting of two components. The first component is made up of small pieces suspended in the second component, and the second component undergoes a change of phase at a prescribed temperature. This phenomenon occurs in a mixture of gravel and wet soil (for example, melting of frozen soil). Existence and uniqueness results of weak solutions of the degenerate parabolic problem are shown by employing monotone operator theory. Local regularity, such as continuity and boundedness of the solution is studied. A discussion is provided about the mutual interplay of the thermodynamic temperature (the temperature in the first component) and the conductive temperature (the temperature in the second component. (Author)
On implicit function theorems and the existence of solutions of nonlinear equations by Hans H Ehrmann(
Book
)
1 edition published in 1962 in English and held by 3 WorldCat member libraries worldwide
Some existence theorems are presented for the solutions of certain nonlinear equations, both local and global theorems. The generality of the local theorems is determined largely by the spaces which contain the domain and the range of the operator in the equation and the elements the operator depends on. (Author)
1 edition published in 1962 in English and held by 3 WorldCat member libraries worldwide
Some existence theorems are presented for the solutions of certain nonlinear equations, both local and global theorems. The generality of the local theorems is determined largely by the spaces which contain the domain and the range of the operator in the equation and the elements the operator depends on. (Author)
On Jointly Estimating Parameters and Missing Data by Maximizing the CompleteData Likelihood by
Roderick J. A Little(
Book
)
2 editions published between 1982 and 1983 in English and held by 3 WorldCat member libraries worldwide
One approach to handling incomplete data occasionally encountered in the literature is to treat the missing data as parameters and to maximize the complete data likelihood over missing data and parameters. This paper points out that although this approach can be useful in particular problems, it is not a generally reliable approach to the analysis of incomplete data. In particular, it does not share the optimal properties of maximum likelihood estimation, except under the trivial asymptotics in which the proportion of missing data goes to zero as the sample size increases. (Author)
2 editions published between 1982 and 1983 in English and held by 3 WorldCat member libraries worldwide
One approach to handling incomplete data occasionally encountered in the literature is to treat the missing data as parameters and to maximize the complete data likelihood over missing data and parameters. This paper points out that although this approach can be useful in particular problems, it is not a generally reliable approach to the analysis of incomplete data. In particular, it does not share the optimal properties of maximum likelihood estimation, except under the trivial asymptotics in which the proportion of missing data goes to zero as the sample size increases. (Author)
A New Class of HermiteBirkhoff Quadrature Formulas of Gaussian Type by
N Dyn(
Book
)
2 editions published between 1979 and 1981 in English and held by 3 WorldCat member libraries worldwide
It is shown how to combine incidence matrices, which admit HermiteBirkhoff quadrature formulas of Gaussian type for any positive measure, in such a way that the resulting matrix also admits Gaussian type quadratures for any positive measure. Moreover, the uniqueness property and the extremal property of the formulas corresponding to the submatrices are transferred to the formula admitted by the composed matrix. (Author)
2 editions published between 1979 and 1981 in English and held by 3 WorldCat member libraries worldwide
It is shown how to combine incidence matrices, which admit HermiteBirkhoff quadrature formulas of Gaussian type for any positive measure, in such a way that the resulting matrix also admits Gaussian type quadratures for any positive measure. Moreover, the uniqueness property and the extremal property of the formulas corresponding to the submatrices are transferred to the formula admitted by the composed matrix. (Author)
Integration of interval functions by
O Caprani(
Book
)
4 editions published between 1980 and 1981 in English and held by 3 WorldCat member libraries worldwide
Caprani, Madsen, and Rall have shown previously that the use of interval values leads to a simple theory of integration in which all functions, interval and real, are integrable. Here, a simplified construction of the interval integral is given for the case that the integrand and interval of integration are finite; the interval integral is shown to be the intersection of the interval Darboux sums corresponding to the partitions of the interval of integration into subintervals of equal length. A rate of convergence of these interval Darboux sums to the interval integral is given for Lipschitz continuous integrands. An alternate approach to interval integration in the unbounded case via finite interval integrals is presented. The results give theoretical support to interval methods for the solution of integral equations and finding extreme values of functionals defined in terms of integrals. (Author)
4 editions published between 1980 and 1981 in English and held by 3 WorldCat member libraries worldwide
Caprani, Madsen, and Rall have shown previously that the use of interval values leads to a simple theory of integration in which all functions, interval and real, are integrable. Here, a simplified construction of the interval integral is given for the case that the integrand and interval of integration are finite; the interval integral is shown to be the intersection of the interval Darboux sums corresponding to the partitions of the interval of integration into subintervals of equal length. A rate of convergence of these interval Darboux sums to the interval integral is given for Lipschitz continuous integrands. An alternate approach to interval integration in the unbounded case via finite interval integrals is presented. The results give theoretical support to interval methods for the solution of integral equations and finding extreme values of functionals defined in terms of integrals. (Author)
A Nonlinear Conservation Law with Memory by
John A Nohel(
Book
)
2 editions published between 1981 and 1982 in English and held by 3 WorldCat member libraries worldwide
In this paper we study a historyboundary value problem for a nonlinear conservation with fading memory in one space dimension. The motivation for studying this problem is an earlier work by C.M. Dafermos and the author concerning the motion of a nonlinear, onedimensional viscoelastic body. Using a variant of an energy method applied to the viscoelastic problem it is shown that under physically reasonable assumptions the nonlinear conservation law has a unique, classical solution (global in time), provided the data are sufficiently smooth and 'small' in a suitable norm; moreover, the solution and its first order derivatives decay to zero as t goes to infinity. The proof illustrates the versatility of the energy method combined with frequency domain techniques for Volterra operators. A preliminary analysis based on current work of R. MalekMadani and the author is presented concerning the development of singularities in smooth solutions of the conservation law (in finite time) for sufficiently 'large' smooth data; under special assumptions it is shown that such singularities necessarily develop. The hope is to apply such a procedure to the viscoelastic problem. (Author)
2 editions published between 1981 and 1982 in English and held by 3 WorldCat member libraries worldwide
In this paper we study a historyboundary value problem for a nonlinear conservation with fading memory in one space dimension. The motivation for studying this problem is an earlier work by C.M. Dafermos and the author concerning the motion of a nonlinear, onedimensional viscoelastic body. Using a variant of an energy method applied to the viscoelastic problem it is shown that under physically reasonable assumptions the nonlinear conservation law has a unique, classical solution (global in time), provided the data are sufficiently smooth and 'small' in a suitable norm; moreover, the solution and its first order derivatives decay to zero as t goes to infinity. The proof illustrates the versatility of the energy method combined with frequency domain techniques for Volterra operators. A preliminary analysis based on current work of R. MalekMadani and the author is presented concerning the development of singularities in smooth solutions of the conservation law (in finite time) for sufficiently 'large' smooth data; under special assumptions it is shown that such singularities necessarily develop. The hope is to apply such a procedure to the viscoelastic problem. (Author)
Convex Solutions to Nonlinear Elliptic and Parabolic Boundary Value Problems by
N. J Korevaar(
Book
)
2 editions published between 1981 and 1983 in English and held by 3 WorldCat member libraries worldwide
This paper contains: (a) A proof that a function on a convex domain whose graph makes zero contact angle with the bounding cylinder and which satisfies an elliptic equation of the appropriate type is convex. (b) A generalization and direct proof of the BrascampLieb result that the first eigenfunction of the Laplacian on a convex domain is Log concave (and so has covex level sets)
2 editions published between 1981 and 1983 in English and held by 3 WorldCat member libraries worldwide
This paper contains: (a) A proof that a function on a convex domain whose graph makes zero contact angle with the bounding cylinder and which satisfies an elliptic equation of the appropriate type is convex. (b) A generalization and direct proof of the BrascampLieb result that the first eigenfunction of the Laplacian on a convex domain is Log concave (and so has covex level sets)
Local Duality of Nonlinear Programs by O Fujiwara(
Book
)
2 editions published between 1982 and 1984 in English and held by 3 WorldCat member libraries worldwide
It is shown that the second order sufficient (necessary) optimality condition for the dual of a nonlinear program is equivalent to the inverse of the Hessian of the Lagrangian being positive definite (semidefinite) on the normal cone to the local primal constraint surface. This compares with the Hessian itself being positive definite (semidefinite) on the tangent cone on the local primal constraint surface for the corresponding second order condition for the primal problem. We also show that primal second order sufficiency (necessity) and dual second order necessity (sufficiency) is essentially equivalent to the Hessian of the Lagrangian being positive definite. This follows from the following interesting linear algebra result: a necessary and sufficient condition for a nonsingular nxn matrix to be positive definite is that for each or some subspace of r(n), the matrix must be positive definite on the subspace and its inverse be positive semidefinite on the orthogonal complement of the subspace. (Author)
2 editions published between 1982 and 1984 in English and held by 3 WorldCat member libraries worldwide
It is shown that the second order sufficient (necessary) optimality condition for the dual of a nonlinear program is equivalent to the inverse of the Hessian of the Lagrangian being positive definite (semidefinite) on the normal cone to the local primal constraint surface. This compares with the Hessian itself being positive definite (semidefinite) on the tangent cone on the local primal constraint surface for the corresponding second order condition for the primal problem. We also show that primal second order sufficiency (necessity) and dual second order necessity (sufficiency) is essentially equivalent to the Hessian of the Lagrangian being positive definite. This follows from the following interesting linear algebra result: a necessary and sufficient condition for a nonsingular nxn matrix to be positive definite is that for each or some subspace of r(n), the matrix must be positive definite on the subspace and its inverse be positive semidefinite on the orthogonal complement of the subspace. (Author)
Stabilization of Solutions of a Degenerate Nonlinear Diffusion Problem by
David Flinker(
Book
)
2 editions published between 1981 and 1982 in English and held by 3 WorldCat member libraries worldwide
2 editions published between 1981 and 1982 in English and held by 3 WorldCat member libraries worldwide
On continued fraction expansions for binomial quadratic surds by
E Frank(
Book
)
2 editions published between 1961 and 1963 in English and held by 2 WorldCat member libraries worldwide
The relations between approximants of the regular continued fraction expansion for square root of C + L and the approximations given by an extension of Newton's formula are given. The general relationship is given between the approximants of a regular periodic continued fraction expansion for square root of C + L and the approximations given by the extended Newton's formula. These formulas are modified to the case of symmetric periodic continued fraction expansions. Rules are given that relate the approximants of mixed periodic continued fraction expansions for square root of C + L to approximations given by the extended Newton's formula. (Author)
2 editions published between 1961 and 1963 in English and held by 2 WorldCat member libraries worldwide
The relations between approximants of the regular continued fraction expansion for square root of C + L and the approximations given by an extension of Newton's formula are given. The general relationship is given between the approximants of a regular periodic continued fraction expansion for square root of C + L and the approximations given by the extended Newton's formula. These formulas are modified to the case of symmetric periodic continued fraction expansions. Rules are given that relate the approximants of mixed periodic continued fraction expansions for square root of C + L to approximations given by the extended Newton's formula. (Author)
The approximate solution of axially symmetric problems: programming the approximate solution of axially symmetric problems(
Book
)
2 editions published in 1963 in English and held by 2 WorldCat member libraries worldwide
2 editions published in 1963 in English and held by 2 WorldCat member libraries worldwide
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