Borel circle squaring

We give a completely constructive solution to Tarski's circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If $k \geq 1$ and $A, B \subseteq \mathbb{R}^k$ are bounded Borel sets with the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than $k$, then $A$ and $B$ are equidecomposable by translations using Borel pieces. This answers a question of Wagon. Our proof uses ideas from the study of flows in graphs, and a recent result of Gao, Jackson, Krohne, and Seward on special types of witnesses to the hyperfiniteness of free Borel actions of $\mathbb{Z}^d$.Comment: Minor typos correcte

Refraction of shear zones in granular materials

We study strain localization in slow shear flow focusing on layered granular materials. A heretofore unknown effect is presented here. We show that shear zones are refracted at material interfaces in analogy with refraction of light beams in optics. This phenomenon can be obtained as a consequence of a recent variational model of shear zones. The predictions of the model are tested and confirmed by 3D discrete element simulations. We found that shear zones follow Snell's law of light refraction.Comment: 4 pages, 3 figures, minor changes, jounal ref. adde

Resolved Spectroscopy of the Narrow-Line Region in NGC 1068: Kinematics of the Ionized Gas

We have determined the radial velocities of the [O III] emitting gas in the inner narrow-line region (NLR) of the Seyfert 2 galaxy NGC 1068, along a slit at position angle of 202 degrees, from STIS observations at a spatial resolution of 0.1 arcsec and a spectral resolving power of approximately 1000. We use these data to investigate the kinematics of the NLR within 6 arcsec (430 pc) of the nucleus. The emission-line knots show evidence for radial acceleration, to a projected angular distance of 1.7 arcsec in most cases, followed by deceleration that approaches the systemic velocity at a projected distance of about 4 arcsec. We find that a simple kinematic model of biconical radial outflow can match the general trend of observed radial velocities. In this model, the emitting material is evacuated along the bicone axis, and the axis is inclined 5 degrees out of the plane of the sky. The acceleration of the emission-line clouds provides support for dynamical models that invoke radiation and/or wind pressure. We suggest that the deceleration of the clouds is due to their collision with a patchy and anistropically distributed ambient medium.Comment: 18 pages, Latex, includes 3 figures in postscript, to appear in the Astrophysical Journal Letter

Measurable realizations of abstract systems of congruences

An abstract system of congruences describes a way of partitioning a space into finitely many pieces satisfying certain congruence relations. Examples of abstract systems of congruences include paradoxical decompositions and $n$-divisibility of actions. We consider the general question of when there are realizations of abstract systems of congruences satisfying various measurability constraints. We completely characterize which abstract systems of congruences can be realized by nonmeager Baire measurable pieces of the sphere under the action of rotations on the $2$-sphere. This answers a question of Wagon. We also construct Borel realizations of abstract systems of congruences for the action of $\mathsf{PSL}_2(\mathbb{Z})$ on $\mathsf{P}^1(\mathbb{R})$. The combinatorial underpinnings of our proof are certain types of decomposition of Borel graphs into paths. We also use these decompositions to obtain some results about measurable unfriendly colorings.Comment: minor correction

Dominant Gender Discourses and Boys’ Interest in Foreign Language Learning

Language learning has long been regarded as a female domain in both commonsense notions about gender and academic research. In fact, there have been numerous studies on girls’ superior performance in language education (Arnot et al. 1996; Burstall 1975; Boyle 1987) and the reasons behind this phenomenon, and more recently, boys have been painted as victims of feminized language learning in education politics (Qualifications and Curriculum Authority 1998). According to poststructuralist research, however, gendered identities are continuously constructed in discourse, which links specific behavior to images of femininity or masculinity. Based on the assumption that these understandings of gender identities play a role in students’ attitudes towards certain practices, this paper will examine the relationship between dominant gender discourses on language learning and boys’ interest in Foreign Language Learning (FLL). For this purpose, poststructuralist gender theory will be applied to popular narratives based on dominant gender discourses, namely sex difference research and male underachievement in language education, in order to show that their perpetuation of the idea that language is a female domain is a possible factor in boys’ lack of FLL interest. The paper aims to illustrate that dominant gender discourses should be problematized as a hindrance to boys’ FLL interest rather than reinforced through educational politics and academic research

Jet triggered Type Ia supernovae in radio-galaxies?

We report the serendipitous discovery of a supernova (SN) in the nearby radio-galaxy 3C 78. Observations obtained with the STIS spectrograph on board the Hubble Space Telescope show, at a distance of 0.54 arcsec (300 pc) from the galaxy nucleus, a second bright source, not present in previous images. As this source was fortuitously covered by the spectrograph slit its spectrum was obtained and it is characteristic of a Type Ia SN. This SN is closely aligned with the radio-jet of 3C 78. Analysis of historical records shows that such a close association between jet and supernova occurred in 6 of the 14 reported SNe in radio-galaxies. The probability that this results from a random distribution of SN in the host galaxy is less than 0.05%. We then argue that jets might trigger supernova explosions.Comment: 9 pages, 3 figures, 1 table, to appear in ApJL, 20 Jul 200

Models relating the radio emission and ionised gas in Seyfert nuclei

Possible models are discussed in which the radio emitting components in Seyfert II nuclei can compress and accelerate the ambient nuclear medium to produce the characteristics of the narrow line region. A first order model, which considers only the expansion of the radio components, is briefly described. However, in many Seyfert nuclei it appears that the linear motion of the radio components is also important. This can result in shock heating of the ambient medium, and if the cooling time is long enough, can lead to a displacement between the radio component and the associated emission lines. This effect may be present in NGC 1068 and NGC 5929 and by considering ram pressure balance and the cooling length it is possible to estimate lobe velocities and ambient densities