RECOVERY OF A SURFACE WITH BOUNDARY AND ITS CONTINUITY AS A FUNCTION OF ITS TWO FUNDAMENTAL FORMS

2005 ◽  
Vol 03 (02) ◽  
pp. 99-117 ◽  
Author(s):  
PHILIPPE G. CIARLET ◽  
CRISTINEL MARDARE
Keyword(s):  
Positive Definite ◽  
Symmetric Matrices ◽  
Partial Derivatives ◽  
Lipschitz Continuous ◽  
Locally Lipschitz ◽  
The Matrix ◽  
Connected Open Subset ◽  
Matrix Fields ◽  
Simply Connected ◽  
Derivatives Of

If a field A of class [Formula: see text] of positive-definite symmetric matrices of order two and a field B of class [Formula: see text] of symmetric matrices of order two satisfy together the Gauss and Codazzi–Mainardi equations in a connected and simply-connected open subset ω of ℝ2, then there exists an immersion [Formula: see text], uniquely determined up to proper isometries in ℝ3, such that A and B are the first and second fundamental forms of the surface θ(ω). Let [Formula: see text] denote the equivalence class of θ modulo proper isometries in ℝ3 and let [Formula: see text] denote the mapping determined in this fashion. The first objective of this paper is to show that, if ω satisfies a certain "geodesic property" (in effect a mild regularity assumption on the boundary of ω) and if the fields A and B and their partial derivatives of order ≤ 2 (respectively, ≤ 1), have continuous extensions to [Formula: see text], the extension of the field A remaining positive-definite on [Formula: see text], then the immersion θ and its partial derivatives of order ≤ 3 also have continuous extensions to [Formula: see text]. The second objective is to show that, if ω satisfies the geodesic property and is bounded, the mapping ℱ can be extended to a mapping that is locally Lipschitz-continuous with respect to the topologies of the Banach spaces [Formula: see text] for the continuous extensions of the matrix fields (A, B), and [Formula: see text] for the continuous extensions of the immersions θ.

C∞-REGULARITY OF A MANIFOLD AS A FUNCTION OF ITS METRIC TENSOR

2006 ◽  
Vol 04 (01) ◽  
pp. 19-30 ◽  
Author(s):  
CRISTINEL MARDARE
Keyword(s):  
Differential Geometry ◽  
Smooth Boundary ◽  
Positive Definite ◽  
Symmetric Matrices ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Metric Tensor ◽  
Basic Theorem ◽  
Connected Open Subset ◽  
Simply Connected

A basic theorem from differential geometry asserts that if the Riemann curvature tensor associated with a smooth field C of positive-definite symmetric matrices of order n vanishes in a simply-connected open subset Ω of ℝn, then C is the metric tensor of a manifold isometrically immersed in ℝn. If Ω is connected, then the isometric immersion Θ defined in this fashion is unique up to isometries of ℝn. We prove that if the set Ω is bounded and has a smooth boundary, then the mapping C ↦ Θ is of class C∞ between manifolds in appropriate Banach spaces.

On regularized distance and related functions

1979 ◽  
Vol 83 (1-2) ◽  
pp. 115-122 ◽  
Author(s):  
L. E. Fraenkel
Keyword(s):  
Continuous Function ◽  
Closed Subset ◽  
Parameter Family ◽  
Lipschitz Continuous Function ◽  
Partial Derivatives ◽  
Lipschitz Continuous ◽  
Order Of Magnitude ◽  
Derivatives Of ◽  
Smooth Approximations

SynopsisLetFbe any closed subset of ℝN. Stein's regularized distance is a smooth (C∞) function, defined on the complementcF, that approximates the distance fromFof any pointx ∈cFin the manner shown by the inequalities (*) in the Introduction below. In this paper we use a method different from Stein's to construct a one-parameter family of smooth approximations to any positive Lipschitz continuous function, with the effect that the constants in (*) can be made arbitrarily close to 1. It is shown that partial derivatives of order two or more, while necessarily unbounded, are best possible in order of magnitude.

scholarly journals Second Order Perturbations of Elliptic Elements with Respect to the Initial Ones

1999 ◽  
Vol 172 ◽  
pp. 453-454
Author(s):  
F.J. Marco Castillo ◽  
M.J. Martínez Usó ◽  
J.A. López Ortí
Keyword(s):  
Quadratic Form ◽  
Second Order ◽  
Initial Instant ◽  
Orbital Elements ◽  
Partial Derivatives ◽  
Orbital Parameters ◽  
The Matrix ◽  
The Individual ◽  
Derivatives Of

AbstractThe following paper is devoted to the theoretical exposition of the obtention of second order perturbations of elliptic elements and is a follow-up of previous papers (Marco et al., 1996; Marco et al., 1997) where the hypothesis was made that the matrix of the partial derivatives of the orbital elements with respect to the initial ones is the identity matrix at the initial instant only. So, we must compute them through the integration of Lagrange planetary equations and their partial derivatives.Such developments have been applied to the individual corrections of orbits together with the correction of the reference system through the minimization of a quadratic form obtained from the linearized residual. In this state two new targets emerged: 1.To be sure that the most suitable quadratic form was to be considered.2.To provide a wider vision of the behavior of the different orbital parameters in time.Both aims may be accomplished through the consideration of the second order partial derivatives of the elliptic orbital elements with respect to the initial ones.

scholarly journals On the estimation of ${x}^TA^{-1}{x}$ for symmetric matrices

2021 ◽  
Vol 37 ◽  
pp. 549-561
Author(s):  
Paraskevi Fika ◽  
Marilena Mitrouli ◽  
Ondrej Turec
Keyword(s):  
Quadratic Form ◽  
Mathematical Problem ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Symmetric Matrices ◽  
Symmetric Positive Definite Matrix ◽  
Matrix Inverse ◽  
Symmetric Positive Definite ◽  
The Matrix

The central mathematical problem studied in this work is the estimation of the quadratic form $x^TA^{-1}x$ for a given symmetric positive definite matrix $A \in \mathbb{R}^{n \times n}$ and vector $x \in \mathbb{R}^n$. Several methods to estimate $x^TA^{-1}x$ without computing the matrix inverse are proposed. The precision of the estimates is analyzed both analytically and numerically.  

THE OSCILLATION PATTERN OF SOLUTIONS TO PARABOLIC EQUATIONS AS TIME GOES TO INFINITY

1999 ◽  
Vol 01 (03) ◽  
pp. 451-466 ◽  
Author(s):  
MYRIAM COMTE ◽  
ALAIN HARAUX
Keyword(s):  
Parabolic Equations ◽  
Bounded Solution ◽  
Connected Components ◽  
Semilinear Parabolic Equation ◽  
Oscillation Pattern ◽  
Lipschitz Continuous ◽  
Locally Lipschitz ◽  
Connected Open Subset ◽  
Continuous Boundary ◽  
Semilinear Parabolic

We consider the semilinear parabolic equation [Formula: see text] where Ω be a bounded, connected open subset of ℝN with a Lipschitz continuous boundary and f is a locally Lipschitz continuous function: ℝ→ℝ. If u is a bounded solution de (1.1) for which the ω-limit set satisfies [Formula: see text] the number of connected components of {x∈Ω; φ(x)≠=0}) is equal to a constant n∞ on ω(u) and there exists T>0 such that for all t≥T, the set {x∈Ω; u(t,x)≠0} has a finite number of connected components precisely equal to n∞.

scholarly journals Partial derivative of matrix functions with respect to a vector variable

2008 ◽  
Vol 30 (4) ◽  
Author(s):  
Nguyen Van Khang
Keyword(s):  
Partial Derivative ◽  
Time Derivative ◽  
Matrix Functions ◽  
Kronecker Products ◽  
Partial Derivatives ◽  
Vector Variable ◽  
Dynamics Of Multibody Systems ◽  
The Matrix ◽  
The Relationship ◽  
Derivatives Of

The partial derivatives of scalar functions and vector functions with respect to a vector variable are defined and used in dynamics of multibody systems. However the partial derivative of matrix functions with respect to a vector variable is also still limited. In this paper firstly the definitions of partial derivatives of scalar functions, vector functions and matrix functions with respect to a vector variable are represented systematically. After an overview of the matrix calculus related to Kronecker products is presented. Two theorems which specify the relationship between the time derivative of a matrix and its partial derivative with respect to a vector, and the partial derivative of product of two matrices with respect to a vector, are then proved.

ON THE RECOVERY AND CONTINUITY OF A SUBMANIFOLD WITH BOUNDARY

2005 ◽  
Vol 03 (02) ◽  
pp. 119-143 ◽  
Author(s):  
MARCELA SZOPOS
Keyword(s):  
Euclidean Space ◽  
Banach Spaces ◽  
Riemannian Geometry ◽  
Existence And Uniqueness ◽  
Analogous Result ◽  
Smoothness Assumption ◽  
Lipschitz Continuous ◽  
Locally Lipschitz ◽  
Simply Connected ◽  
Codazzi Equations

The fundamental theorem of Riemannian geometry asserts that a connected and simply-connected Riemannian space ω of ℝp can be isometrically immersed into the Euclidean space ℝp+q if and only if there exist tensors satisfying the Gauss–Ricci–Codazzi equations, in which case these immersions are uniquely determined up to isometries in ℝp+q. In this fashion, we can define a mapping which associates with these prescribed tensors the reconstructed submanifold. The purpose of this paper is twofold: under a smoothness assumption on the boundary of ω, we first establish an analogous result for the existence and uniqueness of a submanifold "with boundary" and then show that the mapping constructed in this fashion is locally Lipschitz-continuous with respect to the topology of the Banach spaces [Formula: see text].

scholarly journals No Infimum Gap and Normality in Optimal Impulsive Control Under State Constraints

2021 ◽  
Author(s):  
Giovanni Fusco ◽  
Monica Motta
Keyword(s):  
Original Problem ◽  
State Constraints ◽  
Impulsive Control ◽  
Sufficient Conditions ◽  
Continuous Data ◽  
Lipschitz Continuous ◽  
Extended Sense ◽  
Unbounded Controls ◽  
Locally Lipschitz ◽  
Optimal Impulsive Control

AbstractIn this paper we consider an impulsive extension of an optimal control problem with unbounded controls, subject to endpoint and state constraints. We show that the existence of an extended-sense minimizer that is a normal extremal for a constrained Maximum Principle ensures that there is no gap between the infima of the original problem and of its extension. Furthermore, we translate such relation into verifiable sufficient conditions for normality in the form of constraint and endpoint qualifications. Links between existence of an infimum gap and normality in impulsive control have previously been explored for problems without state constraints. This paper establishes such links in the presence of state constraints and of an additional ordinary control, for locally Lipschitz continuous data.

Modeling of Distributed Parameter Processes with Neural Networks

2014 ◽  
Vol 555 ◽  
pp. 530-540
Author(s):  
Honoriu Vălean ◽  
Mihail Abrudean ◽  
Mihaela Ligia Ungureşan ◽  
Iulia Clitan ◽  
Vlad Mureşan
Keyword(s):  
Neural Networks ◽  
Simulation Method ◽  
Distributed Parameter ◽  
Hearth Furnace ◽  
Original Solution ◽  
Steel Pipes ◽  
The Matrix ◽  
The Neural Networks ◽  
Derivatives Of ◽  
Seamless Steel

In this paper an original solution for the modeling of distributed parameter processes using neural networks is presented. The proposed method represents a particular alternative to a very accurate modeling-simulation method for this kind of processes, the method based on the matrix of partial derivatives of the state vector (Mpdx), associated with Taylor series. In order to compare the performances generated by the two methods, a distributed parameter thermal process associated to a rotary hearth furnace (R.H.F) from the technological flow of producing seamless steel pipes is considered. The main similarities and differences between the two methods are highlighted in the paper. The treated solution represents a premise for the usage of the neural networks in the automatic control of the distributed parameter processes domain.

Sign in / Sign up

Export Citation Format

Share Document