The aim of this work is the study of the existence and multiplicity of sign changing nonradial solutions to elliptic boundary value problems on annular domains.
We investigate a class of minimal sequences on a finite alphabet Ak = {1,2,...,k} having (k - 1)n + 1 distinct subwords of length n. These sequences, originally defined by P. Arnoux and G. Rauzy, are a natural generalization of binary Sturmian sequences. We describe two simple combinatorial algorithms for constructing characteristic Arnoux-Rauzy sequences (one of which is new even in the Sturmian case). Arnoux-Rauzy sequences arising from fixed points of primitive morphisms are characterized by an underlying periodic structure. We show that every Arnoux-Rauzy sequence contains arbitrarily large subwords of the form V^2+ε and, in the Sturmian case, arbitrarily large subwords of the form V^3+ε. Finally, we prove that an irrational number whose base b-digit expansion is an Arnoux-Rauzy sequence is transcendental.
The investigation of the multifractal spectrum of the equilibrium measure for a parabolic rational map with a Lipschitz continuous potential, φ, which satisfies sup φ < P(φ) x∈J(T) is conducted. More specifically, the multifractal spectrum or spectrum of singularities, f(α) is studied.
In this paper, we prove, for a certain class of open billiard dynamical systems, the existence of a family of smooth probability measures on the leaves of the dynamical system's unstable manifold. These measures describe the conditional asymptotic behavior of forward trajectories of the system. Furthermore, properties of these families are proven which are germane to the PYC programme for these systems. Strong sufficient conditions for the uniqueness of such families are given which depend upon geometric properties of the system's phase space. In particular, these results hold for a fairly nonrestrictive class of triangular configurations of scatterers.
We study the effects of a deformation via the heat equation on closed, plane curves. We begin with an overview of the theory of curves in R3. In particular, we develop the Frenet-Serret equations for any curve parametrized by arc length. This chapter is followed by an examination of curves in R2, and the resultant adjustment of the Frenet-Serret equations. We then prove the rotation index for closed, plane curves is an integer and for simple, closed, plane curves is ±1. We show that a curve is convex if and only if the curvature does not change sign, and we prove the Isoperimetric Inequality, which gives a bound on the area of a closed curve with fixed length. Finally, we study the deformation of plane curves developed by M. Gage and R. S. Hamilton. We observe that convex curves under deformation remain convex, and simple curves remain simple.
Lehr (1991) proved that, if M(q, r) denotes the set of real numbers whose expansion in base-r is q-automatic i.e., is recognized by an automaton A = (Aq, Ar, ao, δ, φ) (or is the image under a letter to letter morphism of a fixed point of a substitution of constant length q) then M(q, r) is closed under addition and rational multiplication. Similarly if we let M(r) denote the set of real numbers α whose base-r digit expansion is ultimately primitive substitutive, i.e., contains a tail which is the image (under a letter to letter morphism) of a fixed point of a primitive substitution then in an attempt to generalize Lehr's result we show that the set M(r) is closed under multiplication by rational numbers. We also show that M(r) is not closed under addition.
We define and construct bicentric circles with respect to three-sided polygons. Then using inherent properties of these circles, we explore both tangent properties, and areas generated from bicentric circles.
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