Vanishing Scalar Invariant Spacetimes in Higher Dimensions

We study manifolds with Lorentzian signature and prove that all scalar curvature invariants of all orders vanish in a higher-dimensional Lorentzian spacetime if and only if there exists an aligned non-expanding, non-twisting, geodesic null direction along which the Riemann tensor has negative boost order.Comment: final versio

Quasi-Exact Solvability and the direct approach to invariant subspaces

We propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of sl(2). This approach is used here to construct new exactly solvable and quasi-exactly solvable quantum Hamiltonians on the line which are not Lie-algebraic. It is also applied to generate potentials with multiple algebraic sectors. We discuss two illustrative examples of these two applications: an interesting generalization of the Lam\'e potential which posses four algebraic sectors, and a quasi-exactly solvable deformation of the Morse potential which is not Lie-algebraic.Comment: 17 pages, 3 figure

Exceptional orthogonal polynomials and the Darboux transformation

We adapt the notion of the Darboux transformation to the context of polynomial Sturm-Liouville problems. As an application, we characterize the recently described $X_m$ Laguerre polynomials in terms of an isospectral Darboux transformation. We also show that the shape-invariance of these new polynomial families is a direct consequence of the permutability property of the Darboux-Crum transformation.Comment: corrected abstract, added references, minor correction

Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension

We consider time reversal transformations to obtain twofold orthogonal splittings of any tensor on a Lorentzian space of arbitrary dimension n. Applied to the Weyl tensor of a spacetime, this leads to a definition of its electric and magnetic parts relative to an observer (i.e., a unit timelike vector field u), in any n. We study the cases where one of these parts vanishes in particular, i.e., purely electric (PE) or magnetic (PM) spacetimes. We generalize several results from four to higher dimensions and discuss new features of higher dimensions. We prove that the only permitted Weyl types are G, I_i and D, and discuss the possible relation of u with the WANDs; we provide invariant conditions that characterize PE/PM spacetimes, such as Bel-Debever criteria, or constraints on scalar invariants, and connect the PE/PM parts to the kinematic quantities of u; we present conditions under which direct product spacetimes (and certain warps) are PE/PM, which enables us to construct explicit examples. In particular, it is also shown that all static spacetimes are necessarily PE, while stationary spacetimes (e.g., spinning black holes) are in general neither PE nor PM. Ample classes of PE spacetimes exist, but PM solutions are elusive, and we prove that PM Einstein spacetimes of type D do not exist, for any n. Finally, we derive corresponding results for the electric/magnetic parts of the Riemann tensor. This also leads to first examples of PM spacetimes in higher dimensions. We also note in passing that PE/PM Weyl tensors provide examples of minimal tensors, and we make the connection hereof with the recently proved alignment theorem. This in turn sheds new light on classification of the Weyl tensors based on null alignment, providing a further invariant characterization that distinguishes the types G/I/D from the types II/III/N.Comment: 43 pages. v2: new proposition 4.10; some text reshuffled (former sec. 2 is now an appendix); references added; some footnotes cancelled, others incorporated into the main text; some typos fixed and a few more minor changes mad

Generalizations of pp-wave spacetimes in higher dimensions

We shall investigate $D$-dimensional Lorentzian spacetimes in which all of the scalar invariants constructed from the Riemann tensor and its covariant derivatives are zero. These spacetimes are higher-dimensional generalizations of $D$-dimensional pp-wave spacetimes, which have been of interest recently in the context of string theory in curved backgrounds in higher dimensions.Comment: 5 pages, RevTex, to appear in Physical Review

Realizations of Real Low-Dimensional Lie Algebras

Using a new powerful technique based on the notion of megaideal, we construct a complete set of inequivalent realizations of real Lie algebras of dimension no greater than four in vector fields on a space of an arbitrary (finite) number of variables. Our classification amends and essentially generalizes earlier works on the subject. Known results on classification of low-dimensional real Lie algebras, their automorphisms, differentiations, ideals, subalgebras and realizations are reviewed.Comment: LaTeX2e, 39 pages. Essentially exetended version. Misprints in Appendix are correcte

A Randomised Placebo-Controlled Trial of a Traditional Chinese Herbal Formula in the Treatment of Primary Dysmenorrhoea

BACKGROUND: Most traditional Chinese herbal formulas consist of at least four herbs. Four-Agents-Decoction (Si Wu Tang) is a documented eight hundred year old formula containing four herbs and has been widely used to relieve menstrual discomfort in Taiwan. However, no specific effect had been systematically evaluated. We applied Western methodology to assess its effectiveness and safety for primary dysmenorrhoea and to evaluate the compliance and feasibility for a future trial. METHODOLOGY/PRINCIPAL FINDINGS: A randomised, double-blind, placebo-controlled, pilot clinical trial was conducted in an ad hoc clinic setting at a teaching hospital in Taipei, Taiwan. Seventy-eight primary dysmenorrheic young women were enrolled after 326 women with self-reported menstrual discomfort in the Taipei metropolitan area of Taiwan were screened by a questionnaire and subsequently diagnosed by two gynaecologists concurrently with pelvic ultrasonography. A dosage of 15 odorless capsules daily for five days starting from the onset of bleeding or pain was administered. Participants were followed with two to four cycles for an initial washout interval, one to two baseline cycles, three to four treatment cycles, and three follow-up cycles. Study outcome was pain intensity measured by using unmarked horizontal visual analog pain scale in an online daily diary submitted directly by the participants for 5 days starting from the onset of bleeding or pain of each menstrual cycle. Overall-pain was the average pain intensity among days in pain and peak-pain was the maximal single-day pain intensity. At the end of treatment, both the overall-pain and peak-pain decreased in the Four-Agents-Decoction (Si Wu Tang) group and increased in the placebo group; however, the differences between the two groups were not statistically significant. The trends persisted to follow-up phase. Statistically significant differences in both peak-pain and overall-pain appeared in the first follow-up cycle, at which the reduced peak-pain in the Four-Agents-Decoction (Si Wu Tang) group did not differ significantly by treatment length. However, the reduced peak-pain did differ profoundly among women treated for four menstrual cycles (2.69 (2.06) cm, mean (standard deviation), for the 20 women with Four-Agents-Decoction and 4.68 (3.16) for the 22 women with placebo, p = .020.) There was no difference in adverse symptoms between the Four-Agents-Decoction (Si Wu Tang) and placebo groups. CONCLUSION/SIGNIFICANCE: Four-Agents-Decoction (Si Wu Tang) therapy in this pilot post-market clinical trial, while meeting the standards of conventional medicine, showed no statistically significant difference in reducing menstrual pain intensity of primary dysmenorrhoea at the end of treatment. Its use, with our dosage regimen and treatment length, was not associated with adverse reactions. The finding of statistically significant pain-reducing effect in the first follow-up cycle was unexpected and warrants further study. A larger similar trial among primary dysmenorrheic young women with longer treatment phase and multiple batched study products can determine the definitive efficacy of this historically documented formula. TRIAL REGISTRATION: Controlled-Trials.com ISRCTN23374750