The Hamilton-Waterloo Problem with even cycle lengths

The Hamilton-Waterloo Problem HWP$(v;m,n;\alpha,\beta)$ asks for a 2-factorization of the complete graph $K_v$ or $K_v-I$, the complete graph with the edges of a 1-factor removed, into $\alpha$ $C_m$-factors and $\beta$ $C_n$-factors, where $3 \leq m < n$. In the case that $m$ and $n$ are both even, the problem has been solved except possibly when $1 \in \{\alpha,\beta\}$ or when $\alpha$ and $\beta$ are both odd, in which case necessarily $v \equiv 2 \pmod{4}$. In this paper, we develop a new construction that creates factorizations with larger cycles from existing factorizations under certain conditions. This construction enables us to show that there is a solution to HWP$(v;2m,2n;\alpha,\beta)$ for odd $\alpha$ and $\beta$ whenever the obvious necessary conditions hold, except possibly if $\beta=1$; $\beta=3$ and $\gcd(m,n)=1$; $\alpha=1$; or $v=2mn/\gcd(m,n)$. This result almost completely settles the existence problem for even cycles, other than the possible exceptions noted above

IUE observations of oxygen-rich supernova remnants

The IUE observations were used to determine the composition of the ejecta (especially C and Si abundances) and to test models for the ionization and excitation of the ejecta of two oxygen-rich supernova remnants (N132D in the Large Magellanic Cloud and 1E 0102-7219 in the Small Magellanic Cloud). Time-dependent photoionization by the EUV and X-ray radiation from 1E 0102-7219 can qualitatively explain its UV and optical line emission, but the density and ionization structures are complex and prevent a unique model from being specified. Many model parameters are poorly constrained, including the time dependence and shape of the ionizing spectrum. Moreover, the models presented are not self-consistent in that the volumes and densities of the optically emitting gas imply optical depths of order unity in the EUV, but absorption of the ionizing radiation was ignored. It is possible that these shortcomings reflect a more fundamental limitation of the model assumptions. It is assumed that the electron velocity distribution is Maxwellian and that the energy deposited by photoionization heats the electrons directly. The 500 eV electrons produced by the Auger process may excite or ionize other ions before they slow down enough to share their energy with other electrons. Many of the excitations would produce photons that could ionize lower ionization stages

The He II Fowler lines and the O III and N III Bowen fluorescence lines in the symbiotic nova RR Tel

A new measure of reddening (E$_{(B-V)}$$\sim$0.00) has been obtained from the comparison between the observed and the theoretical intensity decrement for 20 emission lines of the \ion{He}{ii} Fowler (n$\to$3) series. This value has been confirmed by the STIS and IUE continuum distribution, and by the value of n$_H$ from the damped profile of the IS H Ly-$\alpha$ line. We have obtained very accurate measurements for about thirty Bowen lines of \ion{O}{iii} and a precise determination of the efficiency in the O1 and O3 excitation channels (18 % and 0.7 %, respectively). The relative \ion{O}{iii} intensities are in good agreement with the predictions by Froese Fischer (1994). A detailed study of the decays from all levels involved in the Bowen mechanism has lead to the detection of two new \ion{O}{iii} Bowen lines near $\lambda$ 2190. High resolution IUE data have shown a nearly linear decline with time, from 1978 to 1995, in the efficiency of the O1 and O3 processes, with a steeper slope for the O3 channel. A detailed study of the \ion{N}{iii} $\lambda$ 4640 lines and of their excitation mechanism has shown that, recombination and continuum fluorescence being ruled out, line fluorescence remains the only viable mechanism to pump the 3d $^2D_{5/2}$ and 3d $^2D_{3/2}$ levels of \ion{N}{iii}. We point out the important role of multiple scattering in the resonance lines of \ion{O}{iii} and \ion{N}{iii} near $\lambda$ 374 and show that the observed \ion{N}{iii} line ratios and intensities can be explained in terms of line fluorescence by the three resonance lines of \ion{O}{iii} at $\lambda$$\lambda$ 374.432, 374.162 and 374.073 under optically thick conditions.Comment: Accepted version for Astronomy and Astrophysic

The Peculiar Type Ic Supernova 1997ef: Another Hypernova

SN 1997ef has been recognized as a peculiar supernova from its light curve and spectral properties. The object was classified as a Type Ic supernova (SN Ic) because its spectra are dominated by broad absorption lines of oxygen and iron, lacking any clear signs of hydrogen or helium line features. The light curve is very different from that of previously known SNe Ic, showing a very broad peak and a slow tail. The strikingly broad line features in the spectra of SN 1997ef, which were also seen in the hypernova SN 1998bw, suggest the interesting possibility that SN 1997ef may also be a hypernova. The light curve and spectra of SN 1997ef were modeled first with a standard SN~Ic model assuming an ordinary kinetic energy of explosion $E_{\rm K} = 10^{51}$ erg. The explosion of a CO star of mass $M_{\rm CO} \approx 6 M_\odot$ gives a reasonably good fit to the light curve but clearly fails to reproduce the broad spectral features. Then, models with larger masses and energies were explored. Both the light curve and the spectra of SN 1997ef are much better reproduced by a C+O star model with $E_{\rm K} =$ 8 \e{51} erg and $M_{\rm CO} = 10 M_\odot$. Therefore, we conclude that SN 1997ef is very likely a hypernova on the basis of its kinetic energy of explosion. Finally, implications for the deviation from spherical symmetry are discussed in an effort to improve the light curve and spectral fits.Comment: "To appear in the Astrophysical Journal, Vol.534 (2000)

High Resolution mid-Infrared Imaging of SN 1987A

Using the Thermal-Region Camera and Spectrograph (T-ReCS) attached to the Gemini South 8m telescope, we have detected and resolved 10 micron emission at the position of the inner equatorial ring (ER) of supernova SN 1987A at day 6067. ``Hot spots'' similar to those found in the optical and near-IR are clearly present. The morphology of the 10 micron emission is globally similar to the morphology at other wavelengths from X-rays to radio. The observed mid-IR flux in the region of SN1987A is probably dominated by emission from dust in the ER. We have also detected the ER at 20 micron at a 4 sigma level. Assuming that thermal dust radiation is the origin of the mid-IR emission, we derive a dust temperature of 180^{+20}_{-10} K, and a dust mass of 1.- 8. 10^{-5} Mo for the ER. Our observations also show a weak detection of the central ejecta at 10 micron. We show that previous bolometric flux estimates (through day 2100) were not significantly contaminated by this newly discovered emission from the ER. If we assume that the energy input comes from radioactive decays only, our measurements together with the current theoretical models set a temperature of 90 leq T leq 100 K and a mass range of 10^{-4} - 2. 10^{-3} Mo for the dust in the ejecta. With such dust temperatures the estimated thermal emission is 9(+/-3) 10^{35} erg s^{-1} from the inner ring, and 1.5 (+/-0.5) 10^{36} erg s^{-1} from the ejecta. Finally, using SN 1987A as a template, we discuss the possible role of supernovae as major sources of dust in the Universe.Comment: aastex502, 14 pages, 4 figures; Accepted for publication in ApJ Content changed: new observations, Referee's comments and suggestion

Constructing uniform 2-factorizations via row-sum matrices: solutions to the Hamilton-Waterloo problem

In this paper, we formally introduce the concept of a row-sum matrix over an arbitrary group $G$. When $G$ is cyclic, these types of matrices have been widely used to build uniform 2-factorizations of small Cayley graphs (or, Cayley subgraphs of blown-up cycles), which themselves factorize complete (equipartite) graphs. Here, we construct row-sum matrices over a class of non-abelian groups, the generalized dihedral groups, and we use them to construct uniform $2$-factorizations that solve infinitely many open cases of the Hamilton-Waterloo problem, thus filling up large parts of the gaps in the spectrum of orders for which such factorizations are known to exist