The Lamplighter Group as a Group Generated by a 2-state Automaton, and its Spectrum

We realize the lamplighter group /2 as a group defined by a 2-state automaton. We study the corresponding action of this group on a binary tree and on its boundary. The final goal is the computation for a special system of generators of the spectrum of the Markov (or the random walk) operator which is [–1,1] in this case and of the spectral measure which is a discrete measure concentrated on a dense countable set of points in [–1,1] (a new effect unseen before for Markovian operators on groups which leads to a counterexample to the Strong Atiyah Conjecture). This is done by the computation of spectra of finite-dimensional approximations of the operator and uses an idea of fractalness in a similar way it was used by Bartholdi and Grigorchuk for the computation of the spectra of some branch groups. We also obtain the asymptotic of type e^–1/1–x of the spectral measure in the neighborhood of 1 and show that Følner sets grow exponentially.     

Decay properties of ground-state and isomer of 103In

The β-decay properties of ground-state and isomer of ¹⁰³In were investigated by means of γ-ray spectroscopy. The half-lives of ^103g In and ^103m In were determined to be 60 ± 1 s and 34 ± 2 s, respectively. Out of 149 γ transitions ascribed to the decay of ^103g In, 104 have been placed in a decay scheme including 55 excited states of ^103g Cd. The main part of the resulting distribution of the Gamow-Teller strength B(GT) is associated with the feeding of ¹⁰³Cd levels at excitation energies around 3 MeV. This observation can be interpreted as a sign of dominant feeding of three-quasiparticle states in ¹⁰³Cd, which correspond to the 1⁺(πg _9/2 ^?1 , vg_7/2)?πg_9/2 shell-model configuration spread over many levels. The sum of the B(GT) values deduced from the present γ-ray data amounts to 0. 34, which provides a lower limit to the total Gamow-Teller strength. The B(GT) distribution for ¹⁰³In decay is compared to the corresponding experimental results for ^105,107,109In and to theoretical predictions for ^99,101In. The latter were calculated by using an advanced spherical shell model. The β-branching ratio for the ^103m In decay is estimated and compared with the relevant values for the neighbouring indium isotopes including ¹⁰¹In whose half-life was determined to be 14. 9 ± 1.2 s.     

I. The properties of hot Ca-like fragments from the 40Ca + 40Ca reaction at 35AMeV

The creation of hot Ca-like fragments was investigated in the ⁴⁰Ca + ⁴⁰Ca reaction at 35 AMeV. Using the AMPHORA 4π detector system, the primary projectile-like fragment was reconstructed and its properties were determined. Both primary and secondary distributions are compared with the predictions of a Monte Carlo code describing a heavy-ion collision as a two-step process. Some of the nucleons which are identified as participants in the first step are transferred in the second step to these final states, which correspond on the average to the maximum value of entropy (thermodynamic probability). The model allows for competition between mean-field effects and nucleon-nucleon interactions in the overlap zone of the interacting nuclei. The analysis presented here suggests a thermalized source picture of the decay of the projectile-like fragment. The validity of the reconstruction procedure for projectile-like fragments is discussed. Received: 12 March 2001 / Accepted: 20 June 2001     

Trigonometric identities in *-algebras

We deal with two systems of trigonometric equations where unknown functions are defined on a group and take values in a (not-necessarily commutative) complex algebra with involution. Our purpose is to give some extension of sine and cosine functions from the scalar to the vector case.     

Limit Cycles Appearing After Perturbation of Certain Multidimensional Vector Fields

Let an unperturbed multidimensional polynomial vector field have an invariant plane L and let the system restricted to this plane be Hamiltonian with a quadratic Hamilton function. Now take a polynomial perturbation of this system. The new system has an invariant surface close to L and the system restricted to it has a certain number of limit cycles. We strive to estimate this number. The linearization of this problem leads to estimation of the number of zeros of certain integral, which is a generalization of the abelian integral. We estimate this number of zeros by C ₁+C ₂ n, where n is the degree of the perturbation.     

Study of Stability of Sesame Oil-In-Water Emulsions Determined using an Optical Analyzer and Measurement of Particle Size and Distribution

The aim of the study was to determine the optimal conditions, that is, the content of sesame oil and the amount of carboxymethylcellulose, to obtain stable dispersion systems. Emulsions were stored under different temperature conditions. For this purpose, six emulsions were prepared and their stability was examined empirically using techniques including particle size analysis, optical microscopy, and the Turbiscan test. The emulsion containing 40 g of oil and 0.6 g of thickener had the highest stability. No destabilizations in whole the range of stored temperature were observed for the emulsion with that composition. Emulsion was stable as well at cool temperature as at extreme environment (40°C). Nowadays, the use of O/W emulsions based on vegetable oil is continuously increasing. To date, sesame oil has been used mainly for direct consumption. The present work indicates a novel possibility for use of sesame oil as the fat base of an emulsion system. Besides, the study gave information about parameters of long-term stability emulsion what is the key in the quality of the dispersion systems. This knowledge is necessary for the industry in order to avoid destabilizing processes occurring in a new product.     

A new mesogenic mixture with antiferroelectric phase only at a broad temperature range

New compounds (S)-4?-(1-methylheptyloxycarbonyl)biphenyl-4-yl 4-[3-(2,2,3,3,4,4,5,5,5-nonafluoropentyloxy)prop-1-oxy]benzoates were synthesised and characterised. Their mesomorphic properties were tested by means of polarising optical microscopy and differential scanning calorimetry. The multicomponent antiferroelectric high tilt mixture with long pitch has been also formulated and characterised. The helical pitch of prepared mixture was estimated with selective reflection method.     

Synthesis and characterization of LnFeO<Subscript>3</Subscript> pigments

This work is focused on determination of the optimum firing temperature which leads to formation of the compounds LnFeO₃ with good pigment-application properties (Ln=Gd, La, Yb, Tm, Lu). Based on results of thermal analysis the compounds were prepared by the solid-state reaction at temperature 900 and 1000°C. Colours of pigments vary depending on type of used lanthanoid (from light sienna to dark brown) and also on the type of precursor. Generally, the compounds prepared from iron oxide have more interesting color properties. Colour of these compounds is brighter and deeper. Increasing of the calcination temperature from 900 to 1000°C causes the darkening of colour. The most problably, the darkening is connected with partial reduction of Fe³⁺ to Fe²⁺.