The relationship of local ordering and long-range order is studied for quasicrystalline tilings of plane and space. Two versions of the concept of local rules are introduced: strong and weak. Necessary conditions of the existence of strong local rules are found. They are mainly reduced to the constraints for irrational numbers related to incommensurabilities of the quasicrystals. For planar qua...

A quadratic function q on an Abelian groupG is a map, with values in an Abelian group, such that the map b : (x, y) 7→ q(x + y) − q(x) − q(y) is Z-bilinear. Such a map q satisfies q(0) = 0. If, in addition, q satisfies the relation q(nx) = nq(x) for all n ∈ Z and x ∈ G, then q is homogeneous. In general, a quadratic function cannot be recovered from the associated bilinear pairing b. Homogeneou...

In this paper, we present the exact Hausdorff distance between the offset curve of quadratic Bézier curve and its quadratic GC1 approximation. To illustrate the formula for the Hausdorff distance, we give an example of the quadratic GC1 approximation of the offset curve of a quadratic Bézier curve. 1. Preliminaries Quadratic Bézier curves and their offset curves are widely used in CAD/CAM or Co...

For R-vector spaces U and V we consider a symmetric bilinear map B : U×U → V . This then defines a quadratic map QB : U → V by QB(u) = B(u, u). Corresponding to each λ ∈ V ∗ is a R-valued quadratic form λQB on U defined by λQB(u) = λ · QB(u). B is definite if there exists λ ∈ V ∗ so that λQB is positive-definite. B is indefinite if for each λ ∈ V ∗, λQB is neither positive nor negative-semidefi...

Article history: Received 6 May 2013 Received in revised form 21 November 2014 Accepted 4 January 2015 Available online 13 January 2015

The inclusion of transaction costs is an essential element of any realistic portfolio optimization. In this paper, we consider an extension of the standard portfolio problem in which transaction costs are incurred to rebalance an investment portfolio. The Markowitz framework of mean-variance efficiency is used with costs modelled as a percentage of the value transacted. Each security in the por...

The present article is devoted to a fine study of the convergence of renormalized weighted quadratic and cubic variations of a fractional Brownian motion B with Hurst index H . In the quadratic (resp. cubic) case, when H < 1/4 (resp. H < 1/6), we show by means of Malliavin calculus that the convergence holds in L toward an explicit limit which only depends on B. This result is somewhat surprisi...

The present article is devoted to a fine study of the convergence of renormalized weighted quadratic and cubic variations of a fractional Brownian motion B with Hurst index H . In the quadratic (resp. cubic) case, when H < 1/4 (resp. H < 1/6), we show by means of Malliavin calculus that the convergence holds in L2 toward an explicit limit which only depends on B. This result is somewhat surpris...