TURTLE GRAPHICS OF MORPHIC SEQUENCES

The simplest infinite sequences that are not ultimately periodic are pure morphic sequences: fixed points of particular morphisms mapping single symbols to strings of symbols. A basic way to visualize a sequence is by a turtle curve: for every alphabet symbol fix an angle, and then consecutively for all sequence elements draw a unit segment and turn the drawing direction by the corresponding angle. This paper investigates turtle curves of pure morphic sequences. In particular, criteria are given for turtle curves being finite (consisting of finitely many segments), and for being fractal or self-similar: it contains an up-scaled copy of itself. Also space-filling turtle curves are considered, and a turtle curve that is dense in the plane. As a particular result we give an exact relationship between the Koch curve and a turtle curve for the Thue–Morse sequence, where until now for such a result only approximations were known.

CONSTRUCTION OF CHAOTIC DYNAMICAL SYSTEM

The first‐order difference equation xn+ 1 = f(xn ), n = 0,1,…, where f: R → R, is referred as an one‐dimensional discrete dynamical system. If function f is a chaotic mapping, then we talk about chaotic dynamical system. Models with chaotic mappings are not predictable in long‐term. In this paper we consider family of chaotic mappings in symbol space S 2. We use the idea of topological semi‐conjugacy and so we can construct a family of mappings in the unit segment such that it is chaotic.

Double cavity flow past a wedge

A mathematical model of supercavitating flow past a wedge with sides of arbitrary length is proposed. The flow branches at a point on the lower side of the wedge. At the vertex of the wedge and at the ends of the wedge, the flow breaks away forming a nose bubble and a trailing cavity. The closure mechanism is described by the Tulin single-spiral-vortex model. The flow domain is mapped into a parametric plane cut along a unit segment. The conformal mapping function is reconstructed through the exact solution of two Riemann–Hilbert problems on a genus-zero Riemann surface. To complete the solution, one needs to determine five real parameters from a certain system of transcendental equations. Numerical results are presented for the case when a wedge can rotate about the vertex in the flow domain. In this case, the flow branches at the vertex and the number of the parameters to be determined is 3.

THREE-PAGE ENCODING AND COMPLEXITY THEORY FOR SPATIAL GRAPHS

A finitely presented semigroup RSGn is constructed for n ≥ 2. The centre of RSGn encodes uniquely up to rigid ambient isotopy in 3-space all nonoriented spatial graphs with vertices of degree ≤ n. This encoding is obtained by using three-page embeddings of graphs into the three-page book T × I, where T is the cone on three points, and I is the unit segment. The notion of the three-page complexity for spatial graphs is introduced via three-page embeddings. This complexity satisfies the properties of finiteness and additivity under natural operations.

MULTIFRACTAL SPECTRUM AND THERMODYNAMICAL FORMALISM OF THE FAREY TREE

Let (Ω, μ) be a set of real numbers to which we associate a measure μ. Let α≥0, let Ωα={x∈Ω/α(x)=α}, where α is the concentration index defined by Halsey et al. [1986]. Let fH(α) be the Hausdorff dimension of Ωα. Let fL(α) be the Legendre spectrum of Ω, as defined in [Riedi & Mandelbrot, 1998]; and fC(α) the classical computational spectrum of Ω, defined in [Halsey et al., 1986]. The task of comparing fH, fC and fL for different measures μ was tackled by several authors [Cawley & Mauldin, 1992; Mandelbrot & Riedi, 1997; Riedi & Mandelbrot, 1998] working, mainly, on self-similar measures μ. The Farey tree partition in the unit segment induces a probability measure μ on an universal class of fractal sets Ω that occur in physics and other disciplines. This measure μ is the Hyperbolic measure μℍ, fundamentally different from any self-similar one. In this paper we compare fH, fC and fL for μℍ.

A Geometrical Approach to the Six Trigonometric Ratios

The sine and cosine ratios are often visualized as lengths of the projections of a unit segment on the x-and y-axis. This type of pictorial approach for the other four trigonometric ratios has been presented in the problems sections of some textbooks (e.g., Wooton, Beckenbach, and Dolciani 1969, p. 82, no. 29; p. 85, no. 27; Vance 1963, p. 70, nos. 14–16). However, relationships among the trigonometric ratios are often thought of as algebraic rather than geometric. This article presents geometric interpretations of the six trigonometric ratios, as well as some trigonometric relationships derived geometrically rather than algebraically.