A note on the Zassenhaus product formula

We provide a simple method for the calculation of the terms c_n in the Zassenhaus product $e^{a+b}=e^a e^b \prod_{n=2}^{\infty} e^{c_n}$ for non-commuting a and b. This method has been implemented in a computer program. Furthermore, we formulate a conjecture on how to translate these results into nested commutators. This conjecture was checked up to order n=17 using a computer

On a generalization of Jacobi's elliptic functions and the Double Sine-Gordon kink chain

A generalization of Jacobi's elliptic functions is introduced as inversions of hyperelliptic integrals. We discuss the special properties of these functions, present addition theorems and give a list of indefinite integrals. As a physical application we show that periodic kink solutions (kink chains) of the double sine-Gordon model can be described in a canonical form in terms of generalized Jacobi functions.Comment: 18 pages, 9 figures, 3 table

Early stages of natural revegetation of metalliferous mine workings in South Central Africa: a preliminary survey

The various types of mining sites resulting from human activities in the Katangan Copper Bow and the Zambian Copperbelt are described and a typology is presented whereby ten different situations are recognized. Performance as well as distribution of the diverse plant species observed on these sites is considered. A set of nine ecological conditions is suggested, based both upon the heavy metal content of soil as well as its state of hydration. One taxon is identified as an indicator of each condition recognized. The information presented here is a preliminary requirement for planning the revegetation of metalliferous sites within the area

Investigation of Field and Laboratory Methods for Evaluating Subgrade Support in the Design of Highway Flexible Pavements

Four different methods of evaluating subgrade support under flexible pavements were studied: (1) Field CBR; (2) North Dakota Cone; (3) Bearing Plates; and (4) Laboratory CBR. Approximately 435 miles of flexible pavements in Kentucky were represented. The roads were selected so as to give a wide range in conditions of traffic, soil areas, and design. A total of 185 locations were investigated, and 338 cone tests, 291 field CBR\u27s, and 117 series of plate tests were conducted. There were 178 subgrade samples for which the laboratory CBR test was conducted. Undisturbed samples for future triaxial tests were obtained at 21 locations. Subgrade moisture variation was considered. Traffic was determined by loadometer surveys and use of traffic flow maps. For the traffic imposed, adequacy of the designs -- as indicated by the presence or absence of base failures was evaluated from the standpoint of subgrade support measured by the four methods of test. Comparisons among the various methods of test in determining the subgrade support were made. The ultimate objective is a design criteria for flexible pavements in Kentucky

Reconstruction of Black Hole Metric Perturbations from Weyl Curvature II: The Regge-Wheeler gauge

Perturbation theory of rotating black holes is described in terms of the Weyl scalars $\psi_4$ and $\psi_0$; each satisfying the Teukolsky's complex master wave equation with spin $s=\mp2$, and respectively representing outgoing and ingoing radiation. We explicitly construct the metric perturbations out of these Weyl scalars in the Regge-Wheeler gauge in the nonrotating limit. We propose a generalization of the Regge-Wheeler gauge for Kerr background in the Newman-Penrose language, and discuss the approach for building up the perturbed spacetime of a rotating black hole. We also provide both-way relationships between waveforms defined in the metric and curvature approaches in the time domain, also known as the (inverse-) Chandrasekhar transformations, generalized to include matter.Comment: 22 pages, no figure

Eigenvalue correlations on Hyperelliptic Riemann surfaces

In this note we compute the functional derivative of the induced charge density, on a thin conductor, consisting of the union of g+1 disjoint intervals, $J:=\cup_{j=1}^{g+1}(a_j,b_j),$ with respect to an external potential. In the context of random matrix theory this object gives the eigenvalue fluctuations of Hermitian random matrix ensembles where the eigenvalue density is supported on J.Comment: latex 2e, seven pages, one figure. To appear in Journal of Physics

Tone-activated, remote, alert communication system

Pocket sized transmitter, frequency modulated by crystal derived tones, with integral loop antenna provides police with easy operating alert signal communicator which uses patrol car radio to relay signal. Communication channels are time shared by several patrol units

Cost impact of a non-invasive, portable device for patient self-administration of chronic migraine in a UK National Health Service setting

Background Chronic migraine (CM) is a neurological disorder associated with substantial disability. Botulinum toxin type A (Botox) is an approved and effective preventive treatment option for adult patients with CM. Transcranial magnetic stimulation (TMS) is an alternative treatment device delivering a brief pre-set magnetic pulse used for self-administration by the patient at home. Despite being available in a risk share scheme TMS is perceived to be more costly in the UK. The objective of this study was to analyse the incremental costs of TMS compared to Botox in refractory CM patients both for a UK individual funding request setting as well as for an average UK specialist center setting. Methods Cost impact results were derived from a decision-tree model simulating treatment pathways over 1 year. Costs were applied from the most recently available UK data sources. Sensitivity analysis was performed for all variables. Results Based on published utilisation data 45.5 % of CM patients would continuously receive Botox over 1 year, whereas 53.7 % of TMS patients would be still on treatment at the end of year one. Total costs of Botox treatment accrue to £2923 in an individual funding request NHS cost setting, whereas TMS treatment results in £1466 in the first year. Applying a time-based NHS cost setting expenditures accrue to £1747 for the Botox treatment and to £1361 for the TMS treatment. In both cost settings variation of cost assumptions did have a minor impact on the cost increment from Botox to TMS. Conclusion The current risk share based remuneration model of TMS allows the UK NHS to reimburse only the cost of those patients experiencing reduction in migraine days resulting in lower costs for treating migraine attacks. Treatment of chronic refractory migraine using TMS implies a substantial cost reduction potential for the management of chronic treatment of refractory migraine patients compared to conventional Botox treatment