China
Africa
Explore chapters and articles related to this topic
Mathematical Preliminaries
Published in R. Ravi, Chemical Engineering Thermodynamics, 2020
In section 0.1, we introduce the notation regarding functions and variables. Section 0.2 discusses the inverse function theorem for a function of a single variable. In section 0.3, the differential of a function of one or many variables is introduced and the various misconceptions associated with the term are addressed. Section 0.4 extends the ideas in section 0.2 to a function of many variables. In section 0.5, the implicit function theorem is introduced and it is pointed out how the inverse function theorem may be regarded as a consequence of it. Section 0.6 contains formulae for the chain rule of differentiation in various forms. The interrelationship between the material in sections 0.4–0.6 is brought out. While a procedure for obtaining the formulae in these sections is outlined, the emphasis is on developing an intuitive ability to write these formulae down by inspection. This is on account of the repeated occurrence of instances in thermodynamics where these formulae are required.
Nonlinear metric regularity on fixed sets
Published in Optimization, 2023
Nguyen Huu Tron, Dao Ngoc Han, Huynh Van Ngai
In mathematics, the study of equations of the form where is a single-valued function from a metric space X to another metric space Y, is an important subject. The crucial questions concerning this equation are that when it has a solution and if it has one then how we can find it? As we knew, the basic condition ensuring this equation to have a solution is the surjectivity of f and the invertibility of f ensures the equation to have only one solution. In the special case, due to the inverse function theorem, when is continuously differentiable at some point and the derivative of f at is invertible, then f is too in a neighbourhood of . In the literature, there are some types of regularity of the single-valued mappings which guarantee the existence of solutions of single-valued equations.
Critical point theory for sparse recovery
Published in Optimization, 2023
In case , we set as Its Jacobian at the origin is regular: since ND1 implies that In virtue of the inverse function theorem, Ω is locally invertible, and we obtain by setting : In case , we set as Its Jacobian at the origin is regular: In virtue of the inverse function theorem, Ω is locally invertible, and we obtain by setting : Since we consider locally around the origin on the set it holds . Hence, we get on : In both cases, we arrive at (7), where the coordinate transformation Ψ is understood as the composition of all previous ones, i. e. . Moreover, under the coordinate transformation Ω the set will be equivariantly transformed in itself. The same thus applies for Ψ.
Finding zeros of Hölder metrically subregular mappings via globally convergent Levenberg–Marquardt methods
Published in Optimization Methods and Software, 2022
Masoud Ahookhosh, Ronan M. T. Fleming, Phan T. Vuong
In order to prove Assertion (iii), let us assume that all accumulation points of are solutions of (1), is an accumulation point such that has full rank, and (80) holds. From the inverse function theorem and the full rank assumption, of , there exists a neighbourhood around 0 such that h is invertible. Therefore, there exists a neighbourhood for such that implying Since is an accumulation point of , contains an infinite number of iteration points of . It remains to show that there exists such that , for all . Hence, for an arbitrary , the set involves only a finite number of iterations of , i.e. there exists such that It follows from (80) that there exists such that Let us set leading to giving the result.