Fiber polytopes for the projections between cyclic polytopes

The cyclic polytope $C(n,d)$ is the convex hull of any $n$ points on the moment curve ${(t,t^2,...,t^d):t \in \reals}$ in $\reals^d$. For $d' >d$, we consider the fiber polytope (in the sense of Billera and Sturmfels) associated to the natural projection of cyclic polytopes $\pi: C(n,d') \to C(n,d)$ which "forgets" the last $d'-d$ coordinates. It is known that this fiber polytope has face lattice indexed by the coherent polytopal subdivisions of $C(n,d)$ which are induced by the map $\pi$. Our main result characterizes the triples $(n,d,d')$ for which the fiber polytope is canonical in either of the following two senses: - all polytopal subdivisions induced by $\pi$ are coherent, - the structure of the fiber polytope does not depend upon the choice of points on the moment curve. We also discuss a new instance with a positive answer to the Generalized Baues Problem, namely that of a projection $\pi:P\to Q$ where $Q$ has only regular subdivisions and $P$ has two more vertices than its dimension.Comment: 28 pages with 1 postscript figur

Improving the Efficiency of Electrical Energy Distribution with Decentralized Electric Heating Control Based on Nash Equilibrium

The widespread use of electricity determines the development of new methods for effective control of electrical energy consumers in the face of changing constraints. A model of a decentralized control system for a group of electric room heaters based on the collective behavior of automatic machines interacting with a random environment with a limited resource distribution is studied.The considered problem differs from the known ones in that the distribution participants are limited in the use of the resource by the “all or nothing” condition. This means that each electric heater at the current time can use a fixed amount of energy resources or refuse it, and the third is not provided. The decision to connect the heaters to the electrical network is made when performing the Nash equilibrium. The Nash equilibrium condition in this work means that the unused power of the electrical network is lower than the power of any heater not connected to the electrical network.The self-organization procedure of a group of electric heaters is studied. A model of a control system for electric heaters has been developed with the task of distributing a limited resource of electrical energy based on Nash equilibrium, using the principles of decentralized control, information technologies for the development and implementation of control actions by a group of heaters. The experiments carried out have confirmed the effectiveness of a decentralized electric heating control system and allow us to recommend it for practical use. It is shown that the proposed approach opens the way to the construction of cost-effective intelligent electric heating system