We derive the first and second variation formulas for exponentially harmonic maps between Finsler manifolds, and prove that there is no non-degenerate stable exponentially harmonic map between a compact convex hypersurface of the Euclidean space and any compact Finsler manifold.

In this paper, we study the stability of Sacks-Uhlenbeck α-harmonic maps from a Finsler manifold to a Riemannian manifold and its applications. Then we find conditions under which any non-constant α−harmonic maps from a compact Finsler manifold to a standard unit sphere S n(n > 2) is unstable.

The article discusses the application of tension field and harmonic maps in image processing. It introduces the concept of (α,f)- harmonic maps, which encompasses both α- harmonic maps and f-harmonic maps, and delves into their characteristics. Furthermore, it presents a Liouville type theorem for this kind of harmonic maps

Increasing harmonic contamination is one of the concerns in power systems that has been more considered. On the other hand, the high cost of compensation can eliminate the possibility of compensation across the network. Regarding the restructuring of power systems, it is possible to provide the compensation cost by contaminating subscribers. In this paper, a new method for controlling the network harmonic level is presented using harmonic pricing. Determining the fair harmonic contribution at the point of common coupling voltage is the base of pricing method. In addition, a new index has been proposed to determine the harmonic contribution of the customer and utility to the harmonic contamination of the point of common coupling. By using this index, an accurate impression of the customer and utility contribution in all of the dominant harmonic levels is obtained. The simulation results on the IEEE 14-bus standard network illustrate the effectiveness of the proposed method for pricing based on fair contribution of customer and network. Also, the results show that by justly determining the harmonic contribution, the customers can be encouraged to reduce or control their harmonic level. Otherwise, by aggregating the penalty fees, harmonic filters can be installed at the desired location to control the network harmonic control.

In the main this paper introduces the concept of chromatic harmonic polynomials denoted, H (G; x) and chromatic harmonic indices denoted, H (G) of a graph G. The new concept is then applied to  nd-ing explicit formula for the minimum (maximum) chromatic harmonic polynomials and the minimum (maximum) chromatic harmonic index of certain graphs. It is also applied to split graphs and certain derivative split graphs.

The harmonic index H (G), of a graph G is defined as the sum of weights 2 (deg (u)+deg (v))-1 of all edges in E (G), where deg (u) denotes the degree of a vertex u in V (G). In this paper we define the harmonic polynomial of G as H(G, X)=SuvÎE(G)2X deg(u)+deg(v)-1, where ò10 H (G, x) dx H (G). We present explicit formula for the values of harmonic polynomial for several families of specific graphs and we find the lower and upper bound for harmonic index in Caterpillars of diameter 4.

We consider four-dimensional Lie groups equipped with left-invariant Einstein Lorentzian metrics. The harmonicity properties of left-invariant vector fields on these spaces are determined. In some cases, all these vector fields are critical points for the energy functional restricted to vector fields. Left-invariant vector fields defining harmonic maps are also classified, and the energy of these vector fields is explicitly calculated.

Penetration of harmonics from downstream low voltage networks and resonance phenomenon has caused the level of harmonics to increase in transmission networks. Compensation of these harmonics at high voltage levels is neither recommended nor cost effective. This is due to the fact that there is no single large non-linear load which is causing these harmonics, yet filtering at low voltage non-linear load sites cannot completely compensate the harmonics and the remaining harmonic at standard levels still flow into the upstream networks. The idea of controlling harmonics at a medium/high voltage level caused by the remaining harmonics flown from low voltage networks and possibly amplified by resonance conditions with the use of series active power filters has been proposed by the authors in a published work. Since, this application of a series active filter was new, it was preferred to assign a new name for such device, i.e. harmonic power flow controller or simply HPFC. This work looks into the control strategies applicable to a HPFC in more details. The shortcomings of the previous methods are now removed resulting in better performance of a HPFC.