Satellite constructions and geometric classification of Brunnian links

We study satellite operations on Brunnian links. First, we find two special satellite operations, both of which can construct infinitely many distinct Brunnian links from almost every Brunnian link. Second, we give a geometric classification theorem for Brunnian links, characterize the companionship graph defined by Budney in [JSJ-decompositions of knot and link complements in [Formula: see text], Enseign. Math. 3 (2005) 319–359], and develop a canonical geometric decomposition, which is simpler than JSJ-decomposition, for Brunnian links. The building blocks of Brunnian links then turn out to be Hopf [Formula: see text]-links, hyperbolic Brunnian links, and hyperbolic Brunnian links in unlink-complements. Third, we define an operation to reduce a Brunnian link in an unlink-complement into a new Brunnian link in [Formula: see text] and point out some phenomena concerning this operation.

Incorporation of Preferences into Supply Chains DEA Efficiency: A Geometric Attribution Approach

Among many applications, several studies using Data Envelopment Analysis (DEA) have examined and studied the efficiency of supply chains. However, the majority of existing approaches dealing with this research area have ignored the important factor of decision makers’ preferences. The main objective of this article is to provide consistent DEA models that allow for efficiency analysis in order to determine the optimal allocation of resources according to these preferences. We propose three cases that are inspired from the geometric decomposition of preference attributions: (1) horizontal attribution, which is when decision makers treat each supply chain as a single non-detachable entity; (2) vertical attribution, which is when decision makers consider supply chains detachable and (3) combined attribution, which is when decision makers concurrently assign weights to the supply chain and to its members. Based on this suggested decomposition, new DEA models are developed, and an illustrative example is applied. The obtained results are relevant and show that DEA is capable of easily incorporating the preferences of decision-makers without resorting to weight restrictions on inputs or outputs.

Geometric decomposition, potential-based representation and integrability of non-linear systems

This paper considers the problem of representing a sufficiently smooth non-linear dynamical [system] as a structured potential-driven system. The proposed method is based on a decomposition of a differential one-form associated to a given vector field into its exact and anti-exact components, and into its co -exact and anti-coexact components. The decomposition method, based on the Hodge decomposition theorem, is rendered constructive by introducing a dual operator to the standard homotopy operator. The dual operator inverts locally the co-differential operator, and is used in the present paper to identify the symplectic structure of the dynamics. Applications of the proposed approach to gradient systems, Hamiltonian systems and generalized Hamiltonian systems are given to illustrate the proposed approach. Finally, integrability conditions for generalized Hamiltonian systems are established using the proposed decomposition.

A Geometric Interpretation of Zonostrophic Instability

The zonostrophic instability that leads to the emergence of zonal jets in barotropic beta-plane turbulence is analyzed through a geometric decomposition of the eddy stress tensor. The stress tensor is visualized by an eddy variance ellipse whose characteristics are related to eddy properties. The tilt of the ellipse principal axis is the tilt of the eddies with respect to the shear, and the eccentricity of the ellipse is related to the eddy anisotropy, and its size is related to the eddy kinetic energy. Changes of these characteristics are directly related to the vorticity fluxes forcing the mean flow. The statistical state dynamics of the turbulent flow closed at second order is employed as it provides an analytic expression for both the zonostrophic instability and the stress tensor. For the linear phase of the instability, the stress tensor is analytically calculated at the stability boundary. For the nonlinear equilibration of the instability the tensor is calculated in the limit of small supercriticality in which the amplitude of the jet velocity follows Ginzburg–Landau dynamics. It is found that, dependent on the characteristics of the forcing, the jet is accelerated either because the jet primarily anisotropizes the eddies so as to produce upgradient fluxes, or because the jet changes the eddy tilt. The instability equilibrates as these changes are partially reversed by the nonlinear jet–eddy dynamics.