Parity of transversals of Latin squares

We introduce a notion of parity for transversals, and use it to show that in Latin squares of order $2 \bmod 4$, the number of transversals is a multiple of 4. We also demonstrate a number of relationships (mostly congruences modulo 4) involving $E_1,\dots, E_n$, where $E_i$ is the number of diagonals of a given Latin square that contain exactly $i$ different symbols. Let $A(i\mid j)$ denote the matrix obtained by deleting row $i$ and column $j$ from a parent matrix $A$. Define $t_{ij}$ to be the number of transversals in $L(i\mid j)$, for some fixed Latin square $L$. We show that $t_{ab}\equiv t_{cd}\bmod2$ for all $a,b,c,d$ and $L$. Also, if $L$ has odd order then the number of transversals of $L$ equals $t_{ab}$ mod 2. We conjecture that $t_{ac} + t_{bc} + t_{ad} + t_{bd} \equiv 0 \bmod 4$ for all $a,b,c,d$. In the course of our investigations we prove several results that could be of interest in other contexts. For example, we show that the number of perfect matchings in a $k$-regular bipartite graph on $2n$ vertices is divisible by $4$ when $n$ is odd and $k\equiv0\bmod 4$. We also show that $${\rm per}\, A(a \mid c)+{\rm per}\, A(b \mid c)+{\rm per}\, A(a \mid d)+{\rm per}\, A(b \mid d) \equiv 0 \bmod 4$$ for all $a,b,c,d$, when $A$ is an integer matrix of odd order with all row and columns sums equal to $k\equiv2\bmod4$

The final two redshifts for radio sources from the equatorial BRL sample

Best, Rottgering and Lehnert (1999, 2000a) defined a new sample of powerful radio sources from the Molonglo Reference Catalogue, for which redshifts were compiled or measured for 177 of the 178 objects. For the final object, MRC1059-010 (3C249), the host galaxy is here identified using near-infrared imaging, and the redshift is determined from VLT spectroscopy. For one other object in the sample, MRC0320+053 (4C05.14), the literature redshift has been questioned: new spectroscopic observations of this object are presented, deriving a corrected redshift. With these two results, the spectroscopic completeness of this sample is now 100%. New redshifts are also presented for PKS0742+10 from the Wall & Peacock 2.7 GHz catalogue, and PKS1336+003 from the Parkes Selected Regions. PKS0742+10 shows a strong neutral hydrogen absorption feature in its Lyman-alpha emission profile.Comment: 4 pages. LaTeX. Accepted for publication in MNRA

Children's acquisition of science terms: does fast mapping work?

About the book: This proceedings contains 99 selected papers from the 8th Conference of the International Association for the Study of Child Language (IASCL) held in Donostia-San Sebastián in the Spanish Basque Country in July 1999. The proceedings includes the plenary addresses by Jean-Paul Bronckart, Brian MacWhinney, and Miquel Siguan. The other 96 papers are organized into sections on bilingualism, discourse, phonology, language disorders, lexicon, morphology, syntax, and signed languages. Several of these sections include symposia with introductions as well as individual papers

The triggering probability of radio-loud AGN: A comparison of high and low excitation radio galaxies in hosts of different colors

Low luminosity radio-loud active galactic nuclei (AGN) are generally found in massive red elliptical galaxies, where they are thought to be powered through gas accretion from their surrounding hot halos in a radiatively inefficient manner. These AGN are often referred to as "low-excitation" radio galaxies (LERGs). When radio-loud AGN are found in galaxies with a young stellar population and active star formation, they are usually high-power radiatively-efficient radio AGN ("high-excitation", HERG). Using a sample of low-redshift radio galaxies identified within the Sloan Digital Sky Survey (SDSS), we determine the fraction of galaxies that host a radio-loud AGN, $f_{RL}$, as a function of host galaxy stellar mass, $M_*$, star formation rate, color (defined by the 4000 \angstrom break strength), radio luminosity and excitation state (HERG/LERG). We find the following: 1. LERGs are predominantly found in red galaxies. 2. The radio-loud AGN fraction of LERGs hosted by galaxies of any color follows a $f^{LE}_{RL} \propto M^{2.5}_*$ power law. 3. The fraction of red galaxies hosting a LERG decreases strongly for increasing radio luminosity. For massive blue galaxies this is not the case. 4. The fraction of green galaxies hosting a LERG is lower than that of either red or blue galaxies, at all radio luminosities. 5. The radio-loud AGN fraction of HERGs hosted by galaxies of any color follows a $f^{HE}_{RL} \propto M^{1.5}_*$ power law. 6. HERGs have a strong preference to be hosted by green or blue galaxies. 7. The fraction of galaxies hosting a HERG shows only a weak dependence on radio luminosity cut. 8. For both HERGs and LERGs, the hosting probability of blue galaxies shows a strong dependence on star formation rate. This is not observed in galaxies of a different color.[abridged]Comment: 7 pages, 6 figure

Detoxification in rehabilitation in England: effective continuity of care or unhappy bedfellows?

There is evidence that residential detoxification alone does not provide satisfactory treatment outcomes and that outcomes are significantly enhanced when clients completing residential detoxification attend rehabilitation services (Gossop, Marsden, Stewart, & Rolfe, 1999; Ghodse, Reynolds, Baldacchino, et al., 2002). One way of increasing the likelihood of this continuity of treatment is by providing detoxification and rehabilitation within the same treatment facility to prevent drop-out, while the client awaits a rehabilitation bed or in the transition process. However, there is little research evidence available on the facilities that offer both medical detoxification and residential rehabilitation. The current study compares self-reported treatment provision in 87 residential rehabilitation services in England, 34 of whom (39.1%) reported that they offered detoxification services within their treatment programmes. Although there were no differences in self-reported treatment philosophies, residential rehabilitation services that offered detoxification were typically of shorter duration overall, had significantly more beds and reported offering more group work than residential rehabilitation services that did not offer detoxification. Outcomes were also different, with twice as many clients discharged on disciplinary grounds from residential rehabilitation services without detoxification facilities. The paper questions the UK classification of residential drug treatment services as either detoxification or rehabilitation and suggests the need for greater research focus on the aims, processes and outcomes of this group of treatment providers