The first discovery of Beta Cephei (BCep) pulsators in the LMC was announced nine years ago. This was a remarkable discovery, since theoretical analyses of pulsational stability had previously predicted that early B main-sequence stars with metallicities lower than Z = 0.01 should not pulsate at all. Following this announcement, and subsequent announcements of the discovery of 92 BCep candidates in the LMC, more detailed studies adopting a variety of opacity calculations and metal mixtures indicated that BCep pulsations could be explained in low-metallicity environments after all. In order to ascertain the nature of these pulsations, multi-colour photometry of sufficient precision is required. We have obtained 4 weeks of UBVI photometry on two fields in the LMC that surround stars which have been identified as strong Beta Cephei candidates from OGLE data. We report on pulsations detected in two stars taken from these fields. 1. Pulsating B stars Early B stars are very likely to become progenitors of supernovae that will produce a neutron star as the core remnant. A precise understanding of early B stars is essential for a precise understanding of the formation and character of the lower-mass population of core-collapse supernovae. The degree of uncertainty in our present descriptions of stellar structure and evolution is illustrated in a recent paper by Pietrzynski et al. [1], where the mass value of a classical Cepheid variable, determined from traditional evolutionary models, was shown to be approximately 20% in error. Equally dramatic improvements in our understanding of the detailed structure and evolutionary timescales of B stars are possible. Such improvements will have a significant impact on our understanding of the formation of supernovae, and also on our understanding of the formation and character of pulsars. Supernovae play an important role in the study of a wide variety of astrophysical and cosmological problems, while pulsars are set to play a key role in the science programmes of MeerKAT and the SKA. Asteroseismology is currently the most powerful tool available to solve many outstanding questions in stellar structure and evolution [2], since the oscillation frequencies of a star depend very sensitively on the details of its structure and on its size. Early B stars show dominant pulsations with periods in the range 3 – 8 hours. Many of the pulsation modes have relatively low amplitudes, requiring extensive observation to raise the signal-to-noise ratio to sufficient levels. Early B stars that display such pulsations are classified as Beta Cephei stars (after the prototype). Two of the most important questions about early B stars that invite an asteroseismological answer are concerned with the instability strip associated with Beta Cephei-type pulsations: i) Why have only a few O-type stars been identified as Beta Cephei-type pulsators? ii) What are the exact locations of the edges of the instability strip? The project being reported on is concerned with the latter of these two questions. 2. Metallicity and pulsation Almost a full century after the original discovery of Beta Cephei-type variability, the driving mechanism for the pulsations was finally tracked down to the presence of a sharp, localised increase in opacity in the temperature zone where the iron-group elements are appropriately ionised. Naturally then, the extent of pulsation mode driving that can occur in an early B star will depend on its metal content. Until fairly recently, theoretical analyses of pulsational stability for early B stars had predicted that BCep pulsations would not occur in stars with metallicities lower than Z = 0.01 [3]. Pigulski and Kolaczkowski [4] announced the first discovery of Beta Cephei (BCep) pulsators in the LMC in 2002. A few years later, Kolaczkowski and Pigulski identified 92 BCep candidates in the LMC [5]. These discoveries were unexpected, since the average metallicity of stars in the LMC is well established at a value of [Fe/H] = -0.03 (with little dispersion around this value [6]), which corresponds to Z = 0.008 [7], lower than the metallicity threshold quoted above. Following the discovery of BCep stars in the LMC, more detailed studies adopting a variety of opacity calculations and metal mixtures indicated that BCep pulsations could be explained in low-metallicity stars after all [8]. 3. The observing project Precise studies of the pulsation properties of the newly-discovered BCep stars in the LMC are required to put the new modifications in the stellar models to the test. Multi-colour photometry across the optical spectrum is the most widely-used technique for identifying the non-radial degree l of the pulsations (an important parameter that needs to be determined). Furthermore, the largest possible number of observations should be obtained, to raise the signal-to-noise ratio of the data high enough that a useful number of pulsation modes can be detected. We have embarked on a multi-year project to meet these demands for a selection of the BCep candidates that have been announced. The first season of observations was conducted on the 1.0-m telescope at the Sutherland site of the South African Astronomical Observatory (SAAO). The first two target fields for the project contain two of the only three BCep stars that have been identified in the literature to date [4]. We conducted UBVRI photometry on these two fields for two fortnights, using the STE4 camera, and covering a total time base of 48.2 days. This allows for a frequency resolution of 0.03 cycles per day in the detection of periodic signals, using the criterion of Loumos and Deeming [9]. Typical examples of the two fields as they were captured through the B filter on the STE4 camera are displayed in Figure 1: Figure 1(a). The field “LMC1”. Figure 1 (b). The field “LMC2”. 4. Results Approximately 200 measurements in each of U.B,V,R and I were obtained on each of the two selected fields. The frames depicted in Figure 1 clearly show that there are dozens of stars in each field that can be explored for pulsation signatures. As a first step to ascertaining the quality of the output from this 4-week observing allocation, we compared our results to those obtained by Pigulski and Kolaczkowski [4] for the two stars that they identified in these fields: OGLE 051841.98−691051.9 (alias LMC1) and OGLE 052809.21−694432.1 (alias LMC2). Table 1 displays frequencies and amplitudes found by Pigulski and Kolaczkowski (abbreviated as PK[4]) in the I filter and in our work in the B filter respectively: Table 1. Comparison of frequency analysis for [4] and for this work. Star PK [4] (I) frequencies (c/d) amplitudes (mmag) Our work: (B) frequenccies (c/d) amplitudes (mmag) LMC1 f1 5.179046 5.1 f2 3.495009 3.9 2.52 (alias?) 14.9 f3 2.005502 4.5 2.01 42.1 f4 1.684015 3.1 1.71 20.0 f5 3.816132 2.3 3.76 27.9

We show that Eulerian and Lagrangian perturbations can be interpreted as finite differences that arise from suitably defined differential operators. These operators lead to exact rather than to approximate perturbation relations and equations. The equations obtained are more general than those usually encountered, and are de facto linear. No approximation is thus needed to linearise them. We also explore the possibility of extending this formalism to the description of stellar pulsations of arbitrary amplitude without using power series expansions.

The mathematical properties of harmonic functions have made Fourier-based algorithms very popular in searches for periodic behaviour in stellar light curves and other astronomical data. The Discrete Fourier Transform and the Lomb-Scargle periodogram have been in common use for many decades, as methods particularly suitable for the non-equally-spaced time data that are typical of astronomical measurements. The "Generalised Lomb-Scargle" or GLS method is a recent refinement of the Lomb-Scargle method. Recent tests of these methods have emphasised the strengths and deficiencies of each. These test results demonstrate that certain methods are distinctly unsuitable for certain types of data and could lead to erroneous conclusions about the types of periodic behaviour present in measured data. Given the rapidly growing body of time-domain data in astronomy and the considerable importance of some of the conclusions that have been made on the basis of these data, the recent developments in the study of period-finding algorithms are significant. This paper reports on a comparative study of the above-mentioned methods applied to simulated stellar photometry compiled for three observing scenarios: space-based observation, ground-based observer-driven campaigns, and ground-based survey programs.

Light curve data from the Kepler satellite on pulsating eclipsing Algol-type binary systems display a peculiar feature: the primary shows preferential excitation of pulsation modes with frequencies resonant with the orbital frequency of the binary system. A proposed explanation of this phenomenon is tidal driving of pulsations by the secondary. This paper presents a preliminary calculation of the effects of linear representations of tides on the pulsation frequencies of a polytropic primary.

We briefly review Mixing Length Theory and comment on its effectiveness in modelling stellar convection.

Following the announcement of potential B star pulsators discovered in OGLE data on the LMC [1], and in ASAS-3 data on Galactic stars [2], various long-term photometric campaigns on a variety of B stars have been conducted by the authors since late 2009. This paper summarises the recent results of these campaigns.

Preliminary results of a. four-week multi-colour photometric campaign on previously identified beta Cephei stars as well as newly-discovered variable stars in two respective LMC fields are presented. Besides the two targeted beta Cephei stars, at least six further presumed B variables are detected. The strongest identified periods appear to lie on the longward end of the galactic beta Cep instability strip, as predicted by model calculations.

The effects of the geometric distortion of a rapidly rotating star on the relative amplitudes of radial velocity and luminosity variations are modelled. The possibility of determining the inclination angle of the rotation axis of a rapidly rotating star, from observations of its radial velocity and luminosity variations, is then explored and the nature of these effects for a model of an early B main-sequence star are presented. It is found that this approach holds sufficient promise that more detailed calculations should be embarked upon.

In this paper, I adapt a recent work by Zahn et al. on the shape of rapidly rotating stars to explore the degree of oblateness for uniform and differential rotation of the star. I also discuss the relation of these results with the classical Roche limit.

Perturbation theory uses Lagrangian techniques that require vector fields to be compared at finitely separated points. This method can be generalised to the strong gravitational field regime in one of two ways, using either covariant or Lie derivatives. In this paper, I argue that those methods based on the Lie derivative are more useful. The Lie derivative provides a clear picture of how the deformation of the fluid flow takes place. It also provides a natural way to discuss large perturbations. I apply this method to some elementary stability problems in the study of stellar structure.

The analysis of the light curves of 48 B-type stars observed by Kepler is presented. Among these are 15 pulsating stars, all of which show low frequencies, characteristic of slowly pulsating B (SPB) stars. Seven of these stars also show a few weak, isolated high frequencies and they could be considered as SPB/β Cephei (β Cep) hybrids. In all cases, the frequency spectra are quite different from what is seen from ground-based observations. We suggest that this is because most of the low frequencies are modes of high degree which are predicted to be unstable in models of mid-B stars. We find that there are non-pulsating stars within the β Cep and SPB instability strips. Apart from the pulsating stars, we can identify stars with frequency groupings similar to what is seen in Be stars but which are not Be stars. The origin of the groupings is not clear, but may be related to rotation. We find periodic variations in other stars which we attribute to proximity effects in binary systems or possibly rotational modulation. We find no evidence for pulsating stars between the cool edge of the SPB and the hot edge of the δ Sct instability strips. None of the stars shows the broad features which can be attributed to stochastically excited modes as recently proposed. Among our sample of B stars are two chemically peculiar stars, one of which is a HgMn star showing rotational modulation in the light curve.

Feynman, in his work on superfluid helium, attempted both qualitative and quantitative descriptions of superfluid helium’s excitation spectrum. We briefly review Feynman’s approach to superfluid helium and assess the applicability of this theory to neutron star interiors. We find that Feynman’s calculations for superfluid helium may indeed be applied to the neutron superfluid in a neutron star.

HD75445 was recently announced by Kochukhov et al. (2009) to be a low-amplitude roAp star, based on spectroscopic measurements. We present putative pulsation frequencies of HD75445 determined from 22 hours of Johnson B photometry obtained in 2008, 2009 and 2010. We present the first photometric periodicities detected in this star. We make a marginal detection of one of Kochukhov et al. (2009)'s spectroscopic periods, along with a range of confidently detected periodicities covering the low-frequency end of the roAp instability spectrum and the high-frequency end of the Delta Scuti instability spectrum.

We have used Lomb-Scargle periodogram analysis and Monte Carlo significance tests to detect periodicities above the 3-sigma level in the Beta Cephei stars V400 Car, V401 Car, V403 Car and V405 Car. These methods produce six previously unreported periodicities in the expected frequency range of excited pulsations: one in V400 Car, three in V401 Car, one in V403 Car and one in V405 Car. One of these six frequencies is significant above the 4-sigma level. We provide statistical significances for all of the periodicities found in these four stars.

Some Pfaffian manifolds admit the construction of an associated Weyl line-bundle in which the lift of the Pfaffian structure defines a 2-form which is basic. We identify the conditions under which this construction is possible, implement it, and investigate some properties of the foliated structure of these special manifolds and of their canonical flows.

We discuss some commonly used methods for determining the significance of peaks in the periodograms of time series. We review methods for constructing the classical significance tests, their corresponding false alarm probability functions and the role played in these by independent random variables and by empirical and theoretical cumulative distribution functions. We discuss the concepts of independent frequencies and oversampling in periodogram analysis. We then compare the results of new Monte Carlo simulations for evenly spaced time series with results obtained previously by other authors, and present the results of Monte Carlo simulations for a specific unevenly spaced time series obtained for V403 Car.

This paper defines basic potentials for contact structure. Contact manifolds that admit a basic potential are shown to have an additional foliated structure of co-dimension 1. The properties of this new foliation, and its relation to the characteristic vector field, are explored.

A calculus for exact symplectic manifolds is presented. It consists of the familiar exterior calculus augmented by a set of operations, algebraic and differential, admitted naturally by the exact symplectic structure of the underlying space. The result is a high level calculus that is particularly well suited to deal with problems on this type of manifold. Natural differential operators admitted by an exact symplectic structure are identified. The results are also applied to the de Rham cohomology of this type of manifold.

A co-symplectic structure on the cotangent bundleT*X of an arbitrary manifoldX is defined, and the notion of associated symplectic and co-symplectic structures is introduced. By way of example, the two-dimensional case is considered in some detail. The general case is investigated, and some implications of these results for polarizations in geometric quantization are considered.