Invariant measures for the horocycle flow on periodic hyperbolic surfaces

We classify the ergodic invariant Radon measures for the horocycle flow on geometrically infinite regular covers of compact hyperbolic surfaces. The method is to establish a bijection between these measures and the positive minimal eigenfunctions of the Laplacian of the surface. Two consequences arise: if the group of deck transformations G is of polynomial growth, then these measures are classified by the homomorphisms from G ₀ to ℝ where G ₀ ≤ G is a nilpotent subgroup of finite index; if the group is of exponential growth, then there may be more than one Radon measure which is invariant under the geodesic flow and the horocycle flow. We also treat regular covers of finite volume surfaces. The first author was supported by NSF grant 0500630. The second author was supported by NSF grant 0400687.     

Asymptotic windings of horocycles

Let Γ be a torsion-free hyperbolic group. We study Γ-limit groups which, unlike the fundamental case in which Γ is free, may not be finitely presentable or geometrically tractable. We define model Γ-limit groups, which always have good geometric properties (in particular, they are always relatively hyperbolic). Given a strict resolution of an arbitrary Γ-limit group L, we canonically construct a strict resolution of a model Γ-limit group, which encodes all homomorphisms L → Γ that factor through the given resolution. We propose this as the correct framework in which to study Γ-limit groups algorithmically. We enumerate all Γ-limit groups in this framework.     

Nonlinear transverse vibrations of a slightly curved beam carrying a concentrated mass

In this study, a slightly curved Euler Bernoulli beam carrying a concentrated mass was handled. The beam was resting on an elastic foundation and simply supported at both ends. Effects of the concentrated mass on nonlin- ear vibrations were investigated. Sinusoidal and parabolic type functions were used as curvature functions. Equations of motion have cubic nonlinearities because of elongations during vibrations. Damping and harmonic excitation terms were added to the equations of motion. Method of mul- tiple scales, a perturbation technique, was used for solving integro-differential equation analytically. Natural frequen- cies were calculated exactly for different mass ratios, mass locations, curvature functions, and linear elastic foundation coefficients. Amplitude-phase modulation equations were found by considering primary resonance case. Effects of nonlinear terms on natural frequencies were calculated. Frequency-amplitude and frequency-response graphs were plotted. Finally effects of concentrated mass and chosen curvature function on nonlinear vibrations were investigated.     

Exchangeable measures for subshifts

Let Ω be a Borel subset of where S is countable. A measure is called exchangeable on Ω, if it is supported on Ω and is invariant under every Borel automorphism of Ω which permutes at most finitely many coordinates. De-Finetti's theorem characterizes these measures when . We apply the ergodic theory of equivalence relations to study the case , and obtain versions of this theorem when Ω is a countable state Markov shift, and when Ω is the collection of beta expansions of real numbers in [0,1] (a non-Markovian constraint).     

The Horocycle Flow and the Laplacian on Hyperbolic Surfaces of Infinite Genus

Consider a complete hyperbolic surface which can be partitioned into countably many pairs of pants whose boundary components have lengths less than some constant. We show that any infinite ergodic invariant Radon measure for the horocycle flow is either supported on a single horocycle associated with a cusp, or corresponds canonically to an extremal positive eigenfunction of the Laplace–Beltrami operator.     

Thermal decomposition of basic lead carbonate

TG and DTA curves of the thermal decomposition of basic lead carbonate, 2PbCO₃ Pb(OH)₂, are presented and compared with the thermogram of normal lead carbonate PbCO₃. The appearance of different intermediate compounds at identical experimental conditions is ascribed to the degree of perfection of the carbonate crystals. Poorly developed basic lead carbonate crystals, formed in the presence of polyglutamic acid, give no recognizable intermediate compounds.     

Linear versus Non-Linear Acquisition of Step-Functions

We address in this article the following two closely related problems. 1. How to represent functions with singularities (up to a prescribed accuracy) in a compact way. 2. How to reconstruct such functions from a small number of measurements. The stress is on a comparison of linear and non-linear approaches. As a model case, we use piecewise-constant functions on [0,1], in particular, the Heaviside jump function ℋ _t =χ _[0,t]. Considered as a curve in the Hilbert space L ²([0,1]) it is completely characterized by the fact that any two its disjoint chords are orthogonal. We reinterpret this fact in a context of step-functions in one or two variables. Next, we study the limitations on representability and reconstruction of piecewise-constant functions by linear and semi-linear methods. Our main tools in this problem are Kolmogorov’s n-width and ε-entropy, as well as Temlyakov’s (N,m)-width. On the positive side, we show that a very accurate non-linear reconstruction is possible. It goes through a solution of certain specific non-linear systems of algebraic equations. We discuss the form of these systems and methods of their solution, stressing their relation to Moment Theory and Complex Analysis. Finally, we informally discuss two problems in Computer Imaging which are parallel to problems 1 and 2 above: compression of still images and video-sequences on one side, and image reconstruction from indirect measurement (for example, in Computer Tomography), on the other. This research was supported by the ISF, Grant No. 304/05, and by the Minerva Foundation.     

The effect of fluoride ion concentration on apatite formation and on its crystal habit

The effect of fluoride ions on the rate of apatite [ca_10(PO4)6(F,OH)₂] formation and on its crystal habit was investigated. It was found that both at high and at low fluoride concentrations the rate formations is retarded. At two optimal concentrations, namely at 3.4 and 5.5 fluoride: apatite excess of the stoichiometric ratio, the rate is the highest. The crystal habit of the apatite formed at high fluoride concentrations is hexagonal plate-like, while the apatite formed at low fluoride concentrations is dendritic or needle-like. The concentrations of fluoride in the crystals were determined by specific fluoride electrode technique.     

Absolute summability of infinite series

It is shown in [4] that if a normal matrix,A satisfies some conditions then |C,1| _k summability implies |A| _k summability wherek≥1. In the present paper, we consider the converse implication.     

Rearrangements of bounded variation sequences

Letbv be the set of all bounded variation sequences. In the present paper we deduce from a theorem of Mears a necessary and sufficient condition for the rearrangement (a _p(k)) to be of bounded variation whenevera _k)∈bv; interestingly it coincides with Pleasants’ criterion for convergence-preserving.