Catálogo de publicaciones - libros
Introduction to Soliton Theory: Applications to Mechanics
Ligia Munteanu Stefania Donescu
Resumen/Descripción – provisto por la editorial
No disponible.
Palabras clave – provistas por la editorial
No disponibles.
Disponibilidad
Institución detectada | Año de publicación | Navegá | Descargá | Solicitá |
---|---|---|---|---|
No detectada | 2005 | SpringerLink |
Información
Tipo de recurso:
libros
ISBN impreso
978-1-4020-2576-1
ISBN electrónico
978-1-4020-2577-8
Editor responsable
Springer Nature
País de edición
Reino Unido
Fecha de publicación
2005
Información sobre derechos de publicación
© Springer Science + Business Media, Inc. 2005
Cobertura temática
Tabla de contenidos
Mathematical Methods
Ligia Munteanu; Stefania Donescu
Toric (or sparse) elimination theory uses combinatorial and discrete geometry to exploit the structure of a given system of algebraic equations. The basic objects are the Newton polytope of a polynomial, the Minkowski sum of a set of convex polytopes, and a mixed polyhedral subdivision of such a Minkowski sum. Different matrices expressing the toric resultant shall be discussed, and effective methods for their construction will be described based on discrete geometric operations, namely the subdivision-based methods and the incremental algorithm. The former allows us to produce Macaulay-type formulae of the toric resultant by determining a matrix minor that divides the determinant in order to yield the precise resultant. Toric resultant matrices exhibit a quasi-Toeplitz structure, which may reduce complexity by almost one order of magnitude in terms of matrix dimension.
We discuss perturbation methods to avoid the vanishing of the matrix determinant, or of the toric resultant itself, when the coefficients, which are initially viewed as generic, take specialized values. This is applied to the problem of implicitizing parametric (hyper)surfaces in the presence of base points. Another important application from geometric modelling concerns the prediction of the support of the implicit equation, based on toric elimination techniques.
Toric resultant matrices reduce the numeric approximation of all common roots of a polynomial system to a problem in numerical linear algebra. In addition to a survey of recent results, this chapter points to open questions regarding the theory and practice of toric elimination methods.
Part 1 - Introduction to Soliton Theory | Pp. 1-52
Some Properties of Nonlinear Equations
Ligia Munteanu; Stefania Donescu
Toric (or sparse) elimination theory uses combinatorial and discrete geometry to exploit the structure of a given system of algebraic equations. The basic objects are the Newton polytope of a polynomial, the Minkowski sum of a set of convex polytopes, and a mixed polyhedral subdivision of such a Minkowski sum. Different matrices expressing the toric resultant shall be discussed, and effective methods for their construction will be described based on discrete geometric operations, namely the subdivision-based methods and the incremental algorithm. The former allows us to produce Macaulay-type formulae of the toric resultant by determining a matrix minor that divides the determinant in order to yield the precise resultant. Toric resultant matrices exhibit a quasi-Toeplitz structure, which may reduce complexity by almost one order of magnitude in terms of matrix dimension.
We discuss perturbation methods to avoid the vanishing of the matrix determinant, or of the toric resultant itself, when the coefficients, which are initially viewed as generic, take specialized values. This is applied to the problem of implicitizing parametric (hyper)surfaces in the presence of base points. Another important application from geometric modelling concerns the prediction of the support of the implicit equation, based on toric elimination techniques.
Toric resultant matrices reduce the numeric approximation of all common roots of a polynomial system to a problem in numerical linear algebra. In addition to a survey of recent results, this chapter points to open questions regarding the theory and practice of toric elimination methods.
Part 1 - Introduction to Soliton Theory | Pp. 53-77
Solitons and Nonlinear Equations
Ligia Munteanu; Stefania Donescu
Toric (or sparse) elimination theory uses combinatorial and discrete geometry to exploit the structure of a given system of algebraic equations. The basic objects are the Newton polytope of a polynomial, the Minkowski sum of a set of convex polytopes, and a mixed polyhedral subdivision of such a Minkowski sum. Different matrices expressing the toric resultant shall be discussed, and effective methods for their construction will be described based on discrete geometric operations, namely the subdivision-based methods and the incremental algorithm. The former allows us to produce Macaulay-type formulae of the toric resultant by determining a matrix minor that divides the determinant in order to yield the precise resultant. Toric resultant matrices exhibit a quasi-Toeplitz structure, which may reduce complexity by almost one order of magnitude in terms of matrix dimension.
We discuss perturbation methods to avoid the vanishing of the matrix determinant, or of the toric resultant itself, when the coefficients, which are initially viewed as generic, take specialized values. This is applied to the problem of implicitizing parametric (hyper)surfaces in the presence of base points. Another important application from geometric modelling concerns the prediction of the support of the implicit equation, based on toric elimination techniques.
Toric resultant matrices reduce the numeric approximation of all common roots of a polynomial system to a problem in numerical linear algebra. In addition to a survey of recent results, this chapter points to open questions regarding the theory and practice of toric elimination methods.
Part 1 - Introduction to Soliton Theory | Pp. 78-120
Statics and Dynamics of the Thin Elastic Rod
Ligia Munteanu; Stefania Donescu
Toric (or sparse) elimination theory uses combinatorial and discrete geometry to exploit the structure of a given system of algebraic equations. The basic objects are the Newton polytope of a polynomial, the Minkowski sum of a set of convex polytopes, and a mixed polyhedral subdivision of such a Minkowski sum. Different matrices expressing the toric resultant shall be discussed, and effective methods for their construction will be described based on discrete geometric operations, namely the subdivision-based methods and the incremental algorithm. The former allows us to produce Macaulay-type formulae of the toric resultant by determining a matrix minor that divides the determinant in order to yield the precise resultant. Toric resultant matrices exhibit a quasi-Toeplitz structure, which may reduce complexity by almost one order of magnitude in terms of matrix dimension.
We discuss perturbation methods to avoid the vanishing of the matrix determinant, or of the toric resultant itself, when the coefficients, which are initially viewed as generic, take specialized values. This is applied to the problem of implicitizing parametric (hyper)surfaces in the presence of base points. Another important application from geometric modelling concerns the prediction of the support of the implicit equation, based on toric elimination techniques.
Toric resultant matrices reduce the numeric approximation of all common roots of a polynomial system to a problem in numerical linear algebra. In addition to a survey of recent results, this chapter points to open questions regarding the theory and practice of toric elimination methods.
Part 2 - Applications to Mechanics | Pp. 121-148
Vibrations of Thin Elastic Rods
Ligia Munteanu; Stefania Donescu
Toric (or sparse) elimination theory uses combinatorial and discrete geometry to exploit the structure of a given system of algebraic equations. The basic objects are the Newton polytope of a polynomial, the Minkowski sum of a set of convex polytopes, and a mixed polyhedral subdivision of such a Minkowski sum. Different matrices expressing the toric resultant shall be discussed, and effective methods for their construction will be described based on discrete geometric operations, namely the subdivision-based methods and the incremental algorithm. The former allows us to produce Macaulay-type formulae of the toric resultant by determining a matrix minor that divides the determinant in order to yield the precise resultant. Toric resultant matrices exhibit a quasi-Toeplitz structure, which may reduce complexity by almost one order of magnitude in terms of matrix dimension.
We discuss perturbation methods to avoid the vanishing of the matrix determinant, or of the toric resultant itself, when the coefficients, which are initially viewed as generic, take specialized values. This is applied to the problem of implicitizing parametric (hyper)surfaces in the presence of base points. Another important application from geometric modelling concerns the prediction of the support of the implicit equation, based on toric elimination techniques.
Toric resultant matrices reduce the numeric approximation of all common roots of a polynomial system to a problem in numerical linear algebra. In addition to a survey of recent results, this chapter points to open questions regarding the theory and practice of toric elimination methods.
Part 2 - Applications to Mechanics | Pp. 149-172
The Coupled Pendulum
Ligia Munteanu; Stefania Donescu
Toric (or sparse) elimination theory uses combinatorial and discrete geometry to exploit the structure of a given system of algebraic equations. The basic objects are the Newton polytope of a polynomial, the Minkowski sum of a set of convex polytopes, and a mixed polyhedral subdivision of such a Minkowski sum. Different matrices expressing the toric resultant shall be discussed, and effective methods for their construction will be described based on discrete geometric operations, namely the subdivision-based methods and the incremental algorithm. The former allows us to produce Macaulay-type formulae of the toric resultant by determining a matrix minor that divides the determinant in order to yield the precise resultant. Toric resultant matrices exhibit a quasi-Toeplitz structure, which may reduce complexity by almost one order of magnitude in terms of matrix dimension.
We discuss perturbation methods to avoid the vanishing of the matrix determinant, or of the toric resultant itself, when the coefficients, which are initially viewed as generic, take specialized values. This is applied to the problem of implicitizing parametric (hyper)surfaces in the presence of base points. Another important application from geometric modelling concerns the prediction of the support of the implicit equation, based on toric elimination techniques.
Toric resultant matrices reduce the numeric approximation of all common roots of a polynomial system to a problem in numerical linear algebra. In addition to a survey of recent results, this chapter points to open questions regarding the theory and practice of toric elimination methods.
Part 2 - Applications to Mechanics | Pp. 173-196
Dynamics of the Left Ventricle
Ligia Munteanu; Stefania Donescu
Toric (or sparse) elimination theory uses combinatorial and discrete geometry to exploit the structure of a given system of algebraic equations. The basic objects are the Newton polytope of a polynomial, the Minkowski sum of a set of convex polytopes, and a mixed polyhedral subdivision of such a Minkowski sum. Different matrices expressing the toric resultant shall be discussed, and effective methods for their construction will be described based on discrete geometric operations, namely the subdivision-based methods and the incremental algorithm. The former allows us to produce Macaulay-type formulae of the toric resultant by determining a matrix minor that divides the determinant in order to yield the precise resultant. Toric resultant matrices exhibit a quasi-Toeplitz structure, which may reduce complexity by almost one order of magnitude in terms of matrix dimension.
We discuss perturbation methods to avoid the vanishing of the matrix determinant, or of the toric resultant itself, when the coefficients, which are initially viewed as generic, take specialized values. This is applied to the problem of implicitizing parametric (hyper)surfaces in the presence of base points. Another important application from geometric modelling concerns the prediction of the support of the implicit equation, based on toric elimination techniques.
Toric resultant matrices reduce the numeric approximation of all common roots of a polynomial system to a problem in numerical linear algebra. In addition to a survey of recent results, this chapter points to open questions regarding the theory and practice of toric elimination methods.
Part 2 - Applications to Mechanics | Pp. 197-219
The Flow of Blood in Arteries
Ligia Munteanu; Stefania Donescu
Toric (or sparse) elimination theory uses combinatorial and discrete geometry to exploit the structure of a given system of algebraic equations. The basic objects are the Newton polytope of a polynomial, the Minkowski sum of a set of convex polytopes, and a mixed polyhedral subdivision of such a Minkowski sum. Different matrices expressing the toric resultant shall be discussed, and effective methods for their construction will be described based on discrete geometric operations, namely the subdivision-based methods and the incremental algorithm. The former allows us to produce Macaulay-type formulae of the toric resultant by determining a matrix minor that divides the determinant in order to yield the precise resultant. Toric resultant matrices exhibit a quasi-Toeplitz structure, which may reduce complexity by almost one order of magnitude in terms of matrix dimension.
We discuss perturbation methods to avoid the vanishing of the matrix determinant, or of the toric resultant itself, when the coefficients, which are initially viewed as generic, take specialized values. This is applied to the problem of implicitizing parametric (hyper)surfaces in the presence of base points. Another important application from geometric modelling concerns the prediction of the support of the implicit equation, based on toric elimination techniques.
Toric resultant matrices reduce the numeric approximation of all common roots of a polynomial system to a problem in numerical linear algebra. In addition to a survey of recent results, this chapter points to open questions regarding the theory and practice of toric elimination methods.
Part 2 - Applications to Mechanics | Pp. 220-241
Intermodal Interaction of Waves
Ligia Munteanu; Stefania Donescu
Toric (or sparse) elimination theory uses combinatorial and discrete geometry to exploit the structure of a given system of algebraic equations. The basic objects are the Newton polytope of a polynomial, the Minkowski sum of a set of convex polytopes, and a mixed polyhedral subdivision of such a Minkowski sum. Different matrices expressing the toric resultant shall be discussed, and effective methods for their construction will be described based on discrete geometric operations, namely the subdivision-based methods and the incremental algorithm. The former allows us to produce Macaulay-type formulae of the toric resultant by determining a matrix minor that divides the determinant in order to yield the precise resultant. Toric resultant matrices exhibit a quasi-Toeplitz structure, which may reduce complexity by almost one order of magnitude in terms of matrix dimension.
We discuss perturbation methods to avoid the vanishing of the matrix determinant, or of the toric resultant itself, when the coefficients, which are initially viewed as generic, take specialized values. This is applied to the problem of implicitizing parametric (hyper)surfaces in the presence of base points. Another important application from geometric modelling concerns the prediction of the support of the implicit equation, based on toric elimination techniques.
Toric resultant matrices reduce the numeric approximation of all common roots of a polynomial system to a problem in numerical linear algebra. In addition to a survey of recent results, this chapter points to open questions regarding the theory and practice of toric elimination methods.
Part 2 - Applications to Mechanics | Pp. 242-272
On the Tzitzeica Surfaces and Some Related Problems
Ligia Munteanu; Stefania Donescu
Toric (or sparse) elimination theory uses combinatorial and discrete geometry to exploit the structure of a given system of algebraic equations. The basic objects are the Newton polytope of a polynomial, the Minkowski sum of a set of convex polytopes, and a mixed polyhedral subdivision of such a Minkowski sum. Different matrices expressing the toric resultant shall be discussed, and effective methods for their construction will be described based on discrete geometric operations, namely the subdivision-based methods and the incremental algorithm. The former allows us to produce Macaulay-type formulae of the toric resultant by determining a matrix minor that divides the determinant in order to yield the precise resultant. Toric resultant matrices exhibit a quasi-Toeplitz structure, which may reduce complexity by almost one order of magnitude in terms of matrix dimension.
We discuss perturbation methods to avoid the vanishing of the matrix determinant, or of the toric resultant itself, when the coefficients, which are initially viewed as generic, take specialized values. This is applied to the problem of implicitizing parametric (hyper)surfaces in the presence of base points. Another important application from geometric modelling concerns the prediction of the support of the implicit equation, based on toric elimination techniques.
Toric resultant matrices reduce the numeric approximation of all common roots of a polynomial system to a problem in numerical linear algebra. In addition to a survey of recent results, this chapter points to open questions regarding the theory and practice of toric elimination methods.
Part 2 - Applications to Mechanics | Pp. 273-297