We propose a simple version of chaotic inflation which leads to a division of the universe into infinitely many open universes with all possible values of Ω from 1 to 0. 1 E-mail: [email protected] Flatness of the universe (Ω = 1) for a long time has been considered as one of the most definite predictions of the inflationary theory. However, what will happen if ten years from now it wi...

Measurement and verification are one of the prime stages in the entire course of geometrical products in new generation of geometrical product specifications (GPS) standard. Like other kinds of form tolerances, flatness error is one of the important characteristics affecting the functionality and quality of machined components; sufficient efforts have long been made to determine the flatness er...

manufacturing of an optical lenses demand a specialized technology and thus needs sophisticated precision manufacturing process. pmma is one of the most used optical polymer for manufacturing of optical lenses, the shaping of which into the required precision and accuracy is a challenging task. the focus of the present paper is to understand the effect of machining parameters on the surface cha...

Shape is a key characteristic to determine the quality of outgoing flat-rolled products in the steel industry. It is greatly influenced by flatness, a feature to describe how the surface of a rolled product approaches a plane. Flatness is of the utmost importance in steelmaking, since it is used by most downstream processes and customers for the acceptance or rejection of rolled products. Flatn...

2 Equivalence and flatness 6 2.1 Control systems as infinite dimensional vector fields . . . . . . . 6 2.2 Equivalence of systems . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Differential Flatness . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Application to motion planning . . . . . . . . . . . . . . . . . . . 13 2.5 Motion planning with singularities . . . . . . . . . . . ....

A Bézier curve segment is generally rendered by subdividing it into a series of disjoint curve subsegments, and then approximating each subsegment by joining its endpoints by a line segment (chord). The maximum transverse deviation of each curve subsegment from the corresponding chord (the achieved flatness) should be no greater than a minimum error value, , called the flatness. The standard te...

Condition $(PWP)$ which was introduced in (Laan, V., {it Pullbacks and flatness properties of acts I}, Commun. Algebra, 29(2) (2001), 829-850), is related to flatness concept of acts over monoids. Golchin and Mohammadzadeh in ({it On Condition $(PWP_E)$}, Southeast Asian Bull. Math., 33 (2009), 245-256) introduced Condition $(PWP_E)$, such that Condition $(PWP)$ implies it, that is, Condition $...

We discuss in some detail the requirements on an early-Universe model that solves the horizon and flatness problems during the epoch of classical cosmology (t ≥ ti ≫ 10 sec). We show that a dynamical resolution of the horizon problem requires superluminal expansion (or very close to it) and that a truly satisfactory resolution of the flatness problem requires entropy production. This implies th...

Differential flatness is a property which substantially reduces the difficulty involved in generating dynamically feasible trajectories for underactuated robotic systems. However, there is a large class of robotic systems that are not differentially flat, and an efficient method for computing dynamically feasible trajectories does not exist. In this paper we introduce a weaker but more general ...

In this paper the following three goals are addressed. The first goal is to study some strong partial differential equations (PDEs) that imply curvature-flatness, in cases of both symmetric and non-symmetric connection. Although curvature-flatness idea classic for connection, our main theorems about flatness solutions completely new, leaving a while point view geometry entering PDEs. second int...