Development and application of energy decoupling index as Cartesian Vector: evidence from world-wide regional data

Economic growth and energy consumed is critical for sustainable global development. In this paper, an extended Vector version of the commonly used Decoupling Index De of energy elasticity to Gross Domestic Product (GDP) and Decoupling Ratio of Energy Intensity of GDP are used to investigate decoupling phenomenon for the period 1990 to 2014 in the main regions of the World. Using Vector properties, this study overcomes some well-known deficiencies of Energy to Growth elasticity Decoupling Index and suggests the Decoupling Angle as a suitable indicator when describing decoupling. The relationships with aggregate and per capita indicators are also examined. A general finding was that in emerging economies, even when moving to “disconnected” states of decoupling, reduced energy rates were paired with reduced growth rates and accelerated growth rates with increased energy consumption rates. This statement raises questions over long-term decoupling of energy consumption from economic growth.

Asymptotic behavior of solutions to difference equations in Banach spaces

We investigate the asymptotic properties of solutions to higher order nonlinear difference equations in Banach spaces. We introduce a new technique based on a vector version of discrete L'Hospital's rule, remainder operator, and the regional topology on the space of all sequences on a given Banach space. We establish sufficient conditions for the existence of solutions with prescribed asymptotic behavior. Moreover, we are dealing with the problem of approximation of solutions. Our technique allows us to control the degree of approximation of solutions.

Positive Solutions for Discontinuous Systems via a Multivalued Vector Version of Krasnosel’skiĭ’s Fixed Point Theorem in Cones

We establish the existence of positive solutions for systems of second–order differential equations with discontinuous nonlinear terms. To this aim, we give a multivalued vector version of Krasnosel’skiĭ’s fixed point theorem in cones which we apply to a regularization of the discontinuous integral operator associated to the differential system. We include several examples to illustrate our theory.

Systems of implicit fractional fuzzy differential equations with nonlocal conditions

In this paper, two types of fixed point theorems are employed to study the solvability of nonlocal problem for implicit fuzzy fractional differential systems under Caputo gH-fractional differentiability in the framework of generalized metric spaces. First of all, we extend Krasnoselskii?s fixed point theorem to the vector version in the generalized metric space of fuzzy numbers. Under the Lipschitz conditions, we use Perov?s fixed point theorem to prove the global existence of the unique mild fuzzy solution in both types (i) and (ii). When the nonlinearity terms are not Lipschitz, we combine Perov?s fixed point theorem with vector version of Krasnoselskii?s fixed point theorem to prove the existence of mild fuzzy solutions. Based on the advantage of vector-valued metrics and convergent matrix, we attain some properties of mild fuzzy solutions such as the boundedness, the attractivity and the Ulam - Hyers stability. Finally, a computational example is presented to demonstrate the effectivity of our main results.

Differential geometry

This chapter presents some elements of differential geometry, the ‘vector’ version of Euclidean geometry in curvilinear coordinates. In doing so, it provides an intrinsic definition of the covariant derivative and establishes a relation between the moving frames attached to a trajectory introduced in Chapter 2 and the moving frames of Cartan associated with curvilinear coordinates. It illustrates a differential framework based on formulas drawn from Chapter 2, before discussing cotangent spaces and differential forms. The chapter then turns to the metric tensor, triads, and frame fields as well as vector fields, form fields, and tensor fields. Finally, it performs some vector calculus.

An intrinsic and exterior form of the Bianchi identities

We give an elegant formulation of the structure equations (of Cartan) and the Bianchi identities in terms of exterior calculus without reference to a particular basis and without the exterior covariant derivative. This approach allows both structure equations and the Bianchi identities to be expressed in terms of forms of arbitrary degree. We demonstrate the relationship with both the conventional vector version of the Bianchi identities and to the exterior covariant derivative approach. Contact manifolds, codimension one foliations and the Cartan form of classical mechanics are studied as examples of its flexibility and utility.