Pathological abelian groups: a friendly example

We show that the group of bounded sequences of elements of $\mathbb{Z}[\sqrt 2]$ is an example of an abelian group with several well known, and not so well known, pathological properties. It appears to be simpler than all previously known examples for some of these properties, and at least simpler to describe for others.Comment: 6 page

Visualizing curved spacetime

I present a way to visualize the concept of curved spacetime. The result is a curved surface with local coordinate systems (Minkowski Systems) living on it, giving the local directions of space and time. Relative to these systems, special relativity holds. The method can be used to visualize gravitational time dilation, the horizon of black holes, and cosmological models. The idea underlying the illustrations is first to specify a field of timelike four-velocities. Then, at every point, one performs a coordinate transformation to a local Minkowski system comoving with the given four-velocity. In the local system, the sign of the spatial part of the metric is flipped to create a new metric of Euclidean signature. The new positive definite metric, called the absolute metric, can be covariantly related to the original Lorentzian metric. For the special case of a 2-dimensional original metric, the absolute metric may be embedded in 3-dimensional Euclidean space as a curved surface.Comment: 15 pages, 20 figure

Mum. Dad. Do you need some help with that? Empowering older Australians in a digital era.

The change to a digital environment for Australian families is more than simply adopting internet connectivity or a mobile phone. Moving from an analog environment and into a digital sphere for many individuals is confronting: the transition requires digital media literacy, that is an understanding of devices, forms of connectivity, installation of devices and how best to use digital connectivity to connect with other family members. In this Australian study the interviewees revealed that tensions occur between middle and older adults as both generations try to understand the effect of the change to a digital environment on each other and navigate the best path that enables communication and connection between family members. This paper will primarily draw on the interviews held with middle adult John and his mother Vera

Infinitely many algebras derived equivalent to a block

We give a construction that in many cases gives a simple way to construct infinite families of algebras that are not Morita equivalent, but are all derived equivalent to the same block algebra of a finite group, and apply it to some small blocks. We make some remarks relating this construction to Donovan's Conjecture and Broue's Abelian Defect Group Conjecture

Equivalences of derived categories for symmetric algebras

We give a characterization of the sets of objects of the derived category of a block of a finite group algebra (or other symmetric algebra) that occur as the set of images of simple modules under an equivalence of derived categories. We give some applications to proving that certain blocks have equivalent derived categories

p-Adic Lifting Problems and Derived Equivalences

For two derived equivalent $k$-algebras $\bar\Lambda$ and $\bar\Gamma$, we introduce a correspondence between \OO-orders reducing to $\bar\Lambda$ and \OO-orders reducing to $\bar\Gamma$. We outline how this may be used to transfer properties like uniqueness (or non-existence) of a lift between $\bar\Lambda$ and $\bar\Gamma$. As an application, we look at tame algebras of dihedral type with two simple modules, where, most notably, we are able to show that among those algebras only the algebras $\mathcal D^{\kappa,0}(2A)$ and $\mathcal D^{\kappa,0}(2B)$ can actually occur as basic algebras of blocks of group rings of finite groups

The Test of Command: McNaughton and Exercise “Spartan,” 4–12 March 1943

The large-scale General Headquarters (GHQ) exercise known as “Spartan,” held in the south of England during March 1943, was a significant event in the history of the Canadian Army in the Second World War. The purpose of “Spartan” was to test the army in the dual tasks of breaking out of an established bridgehead and making the transition to open warfare. As a direct result of shortcomings on the exercise, three Canadian generals lost their commands. Of greatest significance was the eventual relief of General A.G.L. McNaughton as commander of the First Canadian Army in November 1943. During and after “Spartan” the Chief of the Imperial General Staff (CIGS), General Sir Francis Alan Brooke, and the Commander-in-Chief of Home Forces, General Sir Bernard Paget, claimed that McNaughton’s performance proved his incapacity to lead First Canadian Army in the field. In consequence, Brooke and Paget orchestrated his removal and Canadian military historians have generally supported their assessment. However, the considerable criticism directed at McNaughton resulting from “Spartan” has suffered from oversimplification. This article will review McNaughton’s performance during the exercise and assess its role in his relief