An Equivalent Hermitian Hamiltonian for the non-Hermitian -x^4 Potential

The potential -x^4, which is unbounded below on the real line, can give rise to a well-posed bound state problem when x is taken on a contour in the lower-half complex plane. It is then PT-symmetric rather than Hermitian. Nonetheless it has been shown numerically to have a real spectrum, and a proof of reality, involving the correspondence between ordinary differential equations and integral systems, was subsequently constructed for the general class of potentials -(ix)^N. For PT-symmetric but non-Hermitian Hamiltonians the natural PT metric is not positive definite, but a dynamically-defined positive-definite metric can be defined, depending on an operator Q. Further, with the help of this operator an equivalent Hermitian Hamiltonian h can be constructed. This programme has been carried out exactly for a few soluble models, and the first few terms of a perturbative expansion have been found for the potential m^2x^2+igx^3. However, until now, the -x^4 potential has proved intractable. In the present paper we give explicit, closed-form expressions for Q and h, which are made possible by a particular parametrization of the contour in the complex plane on which the problem is defined. This constitutes an explicit proof of the reality of the spectrum. The resulting equivalent Hamiltonian has a potential with a positive quartic term together with a linear term.Comment: New reference [10] added and discussed. Minor typographical correction

On the Path-Integral Derivation of the Anomaly for the Hermitian Equivalent of the Complex PTPT-Symmetric Quartic Hamiltonian

It can be shown using operator techniques that the non-Hermitian $PT$-symmetric quantum mechanical Hamiltonian with a "wrong-sign" quartic potential $-gx^4$ is equivalent to a Hermitian Hamiltonian with a positive quartic potential together with a linear term. A naive derivation of the same result in the path-integral approach misses this linear term. In a recent paper by Bender et al. it was pointed out that this term was in the nature of a parity anomaly and a more careful, discretized treatment of the path integral appeared to reproduce it successfully. However, on re-examination of this derivation we find that a yet more careful treatment is necessary, keeping terms that were ignored in that paper. An alternative, much simpler derivation is given using the additional potential that has been shown to appear whenever a change of variables to curvilinear coordinates is made in a functional integral.Comment: LaTeX, 12 pages, no figure

NucleationTheory for High–Carbon Bainite

There is a great deal of interest in high{carbon (¸ 0:8 wt%) bainite, both in the context of cast irons and in the development of novel very strong and tough steels. In this paper we investigate whether the theory describing the nucleation of bainite is appropriate for this new class of materialsThe authors are grateful to the Engineering and Physical Sciences Research Council for supporting this research.Peer reviewe

Entrando en cintura: implicaciones de política económica sobre la sostenibilidad de la deuda pública en la zona euro.

En este artículo se muestra el marco teórico implícito en la restricción presupuestaria intertemporal del gobierno. A partir de este, se realizan una serie de simulaciones sobre el problema de sostenibilidad de la deuda pública, y los sacrificios fiscales necesarios para juzgar el actual nivel de endeudamiento sostenible en algunos países de la zona Euro. En particular, Grecia, Portugal, Italia y España deben generar un superávit primario de 44.1%, 10.5%, 6.8% y 4.5% del PIB, respectivamente para mantener sus actuales niveles de endeudamiento. La situación de Reino unido, Alemania y Francia es diferente, puesto que pueden presentar déficits debido a la relación entre crecimiento esperado y tasa de interés

Bimodal Size-distribution of Bainite Plates

There are two well-known phenomena associated with the bainite reaction, which have been exploited in the present work to enhance the mechanical behaviour of steel. Firstly, the bainite plate size decreases as the transformation temperature is reduced. Secondly, it is bad to have large regions of untransformed austenite in the microstructure; this is because they can transform, under the influence of external stress, into corresponding large regions of untempered, brittle martensite. By adopting a two stage heat treatment in which coarse bainite is produced by isothermal transformation at a high temperature, followed by isothermal transformation at a lower temperature, it has been possible to eliminate blocks of austenite. This leads to a microstructure containing an organized dispersion of fine plates of bainitic ferrite in the regions between the coarse plates. The mechanical properties of this mixture are shown to be better than those of bainite obtained by transformation at any single temperature. The experiments have been conducted in the context of very strong steels, where the strength and hardness can exceed 2.5GPa and 650 HV respectively.Peer reviewe

Skeletal Muscle Mass Indices in Healthy Young Mexican Adults Aged 20–40 Years: Implications for Diagnoses of Sarcopenia in the Elderly Population

Background and Objectives. Skeletal muscle and skeletal muscle indices in young adults from developing countries are sparse. Indices and the corresponding cut-off points can be a reference for diagnoses of sarcopenia. This study assessed skeletal muscle using dual-energy X-ray absorptiometry (DXA) in healthy male and female subjects aged 20–40 years and compared their appendicular skeletal muscle mass (ASM) and total-body skeletal muscle (TBSM) indices using certain cut-off points published in the literature. Methods. A sample of 216 healthy adults men and women from northwest Mexico was included. Body composition was assessed by DXA and several published DXA-derived skeletal muscle indices were compared. Results. Both, ASM and TBSM were higher in men compared to the women group (23.0 ± 3.4 versus 15.9 ± 1.6 kg; P<0.05 and 26.5 ± 4.1 versus 16.9 ± 1.9 kg; P<0.05, resp.). These differences were also valid for both indices. When derived cut-off points were compared with the most reported indices, significant differences were found. Interpretation and Conclusion. Published cut-off points from Caucasians are higher than cut-off point derived in this sample of Mexican subjects. The new DXA-derived cut-off points for ASM proposed herein may improve diagnoses of sarcopenia in the geriatric Mexican population

More people, more food, worse water? a global review of water pollution from agriculture

Current patterns of agricultural expansion and intensification are bringing unprecedented environmental externalities, including impacts on water quality. While water pollution is slowly starting to receive the attention it deserves, the contribution of agriculture to this problem has not yet received sufficient consideration. We need a much better understanding of the causes and effects of agricultural water pollution as well as effective means to prevent and remedy the problem. In the existing literature, information on water pollution from agriculture is highly dispersed. This repost is a comprehensive review and covers different agricultural sectors (including crops, livestock and aquaculture), and examines the drivers of water pollution in these sectors as well as the resulting pressures and changes in water bodies, the associated impacts on human health and the environment, and the responses needed to prevent pollution and mitigate its risks