Fakhruddin has proved that for two lattice polygons P and Q any lattice point in their Minkowski sum can be written as a sum of a lattice point in P and one in Q, provided P is smooth and the normal fan of P is a subdivision of the normal fan of Q. We give a shorter combinatorial proof of this fact that does not need the smoothness assumption on P .

A polyhex can be defined as a graph H which is obtained from the hexagonal lattice L6 of the plane by taking a cycle C in the graph of L6 and defining H to be the part of L6 in the disk bounded by C, including C. Similarly, a mono-qpolyhex, q an integer greater than 3, is obtained from the mono-q-hexagonal lattice (all faces are hexagons except one which is a q-gon, and three faces meet at each...

Nanoparticles of Mn0.5Zn0.5-xCdxFe2O4 with x varying from x = 0.0 to 0.3 were prepared by wet chemical co-precipitation method. The structural and magnetic properties were studied by using X-Ray Diffraction (XRD), Scanning Electron Microscope (SEM), Transmission Electron Microscopy (TEM) and Electron Paramagnetic Resonance (EPR) technique. ...

Let Q(S) be the moduli space of area one holomorphic quadratic differentials for an oriented surface S of genus g ≥ 0 with m ≥ 0 punctures and 3g − 3 + m ≥ 2. We show that for every compact subset K of Q(S) the as-ymptotic growth rate δ(K) of the number of periodic orbits of the Teichmüller flow Φ t which are contained in K is not bigger than h = 6g − 6 + 2m, and sup K δ(K) = h. Similarly, h is...

Hence, q is bounded from above and limt→∞ q(t) = q∞ <∞. Note that the function r may change sign. By a solution of () we mean a function x differentiable up to order n which satisfies () on [Tx,∞), Tx ≥ . A solution of () is said to be proper if sup{|x(t)| : t ≥ T} >  for any T ≥ Tx. As usual, a solution x of () is said to be oscillatory if x changes sign for large t. The aim of this pape...

Let Q(S) be the moduli space of area one holomorphic quadratic differentials for an oriented surface S of genus g ≥ 0 with m ≥ 0 punctures and 3g − 3 + m ≥ 2. We show that for every compact subset K of Q(S) the as-ymptotic growth rate δ(K) of the number of periodic orbits of the Teichmüller flow Φ t which are contained in K is not bigger than h = 6g − 6 + 2m, and sup K δ(K) = h. Similarly, h is...

Abstract Let $$G \subsetneq \mathbb {C}$$ G ⊊ C be a bounded, simply connected domain in $$\mathbb , and denote by $$\begin{aligned} Q(G):= \left\{ f: G \longrightarrow \ |\ f \text { is quasiconformal mapping of } onto \right\} \end{aligned}$$ <mml:mta...

In this paper we use concepts from the lattice-based theory of morphological operators and fuzzy sets to develop generalized lattice image operators that are nonlinear convolutions that can be expressed as supremum (resp. infimum) of fuzzy intersection (resp. union) norms. Our emphasis and differences with many previous works is the construction of pairs of fuzzy dilation (sup of fuzzy intersec...

1. STOKES EQUATIONS In this section, we shall study the well posedness of the weak formulation of the steadystate Stokes equations −μ∆u +∇p = f , (1) −divu = 0, (2) where u can be interpreted as the velocity field of an incompressible fluid motion, and p is then the associated pressure, the constant μ is the viscosity coefficient of the fluid. For simplicity, we consider homogenous Dirichlet bo...