Abstract PQ-type adjacency polytopes $$\nabla ^PQ _G$$ ∇ P Q G are lattice polytopes arising from finite graphs G. There is a connection between $$\nabla ^PQ _G$$ ∇ P Q G and the engineering problem known as power-flow study, which models the balance of electric power on a network of power generation. In particular, the normalized volume of $$\nabla ^PQ _G$$ ∇ P Q G plays a central role. In the present paper, we focus on the case where G is a join graph. In particular, formulas of the $$h^*$$ h ∗ -polynomial and the normalized volume of $$\nabla ^PQ _G$$ ∇ P Q G of a join graph G are presented. Moreover, we give explicit formulas of the $$h^*$$ h ∗ -polynomial and the normalized volume of $$\nabla ^PQ _G$$ ∇ P Q G when G is a complete multipartite graph or a wheel graph.

Abstract. In this paper, we discuss the integer decomposition property for Cayley sums and Minkowski sums of lattice polytopes. In fact, we characterize when Cayley sums have the integer decomposition property in terms of Minkowski sums. Moreover, by using this characterization, we consider when Cayley sums and Minkowski sums of [Formula: see text]-convex-normal lattice polytopes have the integer decomposition property. Finally, we also discuss the level property for Minkowski sums and Cayley sums.

Abstract A lattice polytope $$\mathscr {P} \subset \mathbb {R}^d$$ P⊂Rd is called a locally anti-blocking polytope if for any closed orthant $${\mathbb R}^d_{\varepsilon }$$ Rεd in $$\mathbb {R}^d$$ Rd , $$\mathscr {P} \cap \mathbb {R}^d_{\varepsilon }$$ P∩Rεd is unimodularly equivalent to an anti-blocking polytope by reflections of coordinate hyperplanes. We give a formula for the $$h^*$$ h∗ -polynomials of locally anti-blocking lattice polytopes. In particular, we discuss the $$\gamma $$ γ -positivity of $$h^*$$ h∗ -polynomials of locally anti-blocking reflexive polytopes.

Since Stanley's [ Discrete Comput. Geom., 1 (1986), pp. 9--23] introduction of order polytopes, their geometry has been widely used to examine (algebraic) properties of finite posets. In this paper, we follow this route to examine the levelness property of order polytopes, a property generalizing Gorensteinness. This property has been recently characterized by Miyazaki [ J. Algebra, 480 (2017), pp. 215--236] for the case of order polytopes. We provide an alternative characterization using weighted digraphs. Using this characterization, we give a new infinite family of level posets and show that determining levelness is in $\operatorname{co-NP}$. Moreover, we show how a necessary condition of levelness of [ J. Algebra, 431 (2015), pp. 138--161] can be restated in terms of digraphs. We then turn to the more general family of alcoved polytopes. We give a characterization for levelness of alcoved polytopes using the Minkowski sum. Then we study several cases when the product of two polytopes is level. In particular, we provide an example where the product of two level polytopes is not level.

This paper proposes a localization method to estimate position and azimuth of a lunar rover. In the proposed method, the position is precisely estimated by integration of an absolute and a relative position. The absolute position is measured by observing the sun and the earth, and the relative position is determined by dead reckoning. Effectiveness is confirmed by the results of simulations. Moreover, differences between proposed method and Extended Kalman Filter are shown by results of long-distance simulation.