A simple construction of complex equiangular lines

A set of vectors of equal norm in $\mathbb{C}^d$ represents equiangular lines if the magnitudes of the inner product of every pair of distinct vectors in the set are equal. The maximum size of such a set is $d^2$, and it is conjectured that sets of this maximum size exist in $\mathbb{C}^d$ for every $d \geq 2$. We describe a new construction for maximum-sized sets of equiangular lines, exposing a previously unrecognized connection with Hadamard matrices. The construction produces a maximum-sized set of equiangular lines in dimensions 2, 3 and 8.Comment: 11 pages; minor revisions and comments added in section 1 describing a link to previously known results; correction to Theorem 1 and updates to reference

Effects of Forest Fuels Reduction on Small Mammals in the Lake Tahoe Basin

Long-standing policies of fire suppression, common throughout the Western United States, have resulted in unnaturally high fuel loads within the Lake Tahoe Basin. The swollen reservoir of fuel from downed wood, brush, small trees and other large woody plants poses a danger to the forest by increasing the intensity of wildfires. Fuels reduction and forest restoration treatments are used to thin the forest and remove excess ground fuels to mitigate the risk of catastrophic wildfire. To investigate the impact of two types of these treatments on the small mammal community in the Lake Tahoe Basin, a before-after control-impact study was designed and surveys of forest structure and small mammals conducted prior to and after the completion of treatment activities at six sites to the west and northwest of Lake Tahoe. I found that the small mammal community does not seem to respond strongly to treatment, though the intensity of treatment does differentially affect abundance of some species. This may bode well for small mammal communities in the Sierra Nevada as larger areas of forest continue to be treated, but long-term data is needed to better assess small mammal responses to treatment activities that may not have been detected by our short-term study

Recent Development and Results with the MERLIN Tracking Code

MERLIN is a high performance accelerator simulation code which is used for modelling the collimation system at the LHC. It is written in extensible object-oriented C++ so new physics processes can be easily added. In this article we present recent developments needed for the Hi-Lumi LHC and future high energy colliders including FCC, such as hollow electron lenses and composite materials. We also give an overview of recent simulation work, validation against LHC data from run 1 and 2, and loss maps for Hi-Lumi LHC

SIC~POVMs and Clifford groups in prime dimensions

We show that in prime dimensions not equal to three, each group covariant symmetric informationally complete positive operator valued measure (SIC~POVM) is covariant with respect to a unique Heisenberg--Weyl (HW) group. Moreover, the symmetry group of the SIC~POVM is a subgroup of the Clifford group. Hence, two SIC~POVMs covariant with respect to the HW group are unitarily or antiunitarily equivalent if and only if they are on the same orbit of the extended Clifford group. In dimension three, each group covariant SIC~POVM may be covariant with respect to three or nine HW groups, and the symmetry group of the SIC~POVM is a subgroup of at least one of the Clifford groups of these HW groups respectively. There may exist two or three orbits of equivalent SIC~POVMs for each group covariant SIC~POVM, depending on the order of its symmetry group. We then establish a complete equivalence relation among group covariant SIC~POVMs in dimension three, and classify inequivalent ones according to the geometric phases associated with fiducial vectors. Finally, we uncover additional SIC~POVMs by regrouping of the fiducial vectors from different SIC~POVMs which may or may not be on the same orbit of the extended Clifford group.Comment: 30 pages, 1 figure, section 4 revised and extended, published in J. Phys. A: Math. Theor. 43, 305305 (2010

From SICs and MUBs to Eddington

This is a survey of some very old knowledge about Mutually Unbiased Bases (MUB) and Symmetric Informationally Complete POVMs (SIC). In prime dimensions the former are closely tied to an elliptic normal curve symmetric under the Heisenberg group, while the latter are believed to be orbits under the Heisenberg group in all dimensions. In dimensions 3 and 4 the SICs are understandable in terms of elliptic curves, but a general statement escapes us. The geometry of the SICs in 3 and 4 dimensions is discussed in some detail.Comment: 12 pages; from the Festschrift for Tony Sudber

The role of religion in al-Qaeda’s violence

This chapter explores the role of religion in jihadist violence by applying recent insights from the field of religious studies to the case of al-Qaeda. Whereas religion is often perceived as an explanatory factor in al-Qaeda’s attacks, the chapter demonstrates that the meanings attributed to religion are dependent on particular historical and cultural contexts. As a result, the chapter argues, the concept of religion cannot be consistently separated from its alleged secular counterpart, which makes any distinction between religious violence and secular violence arbitrary. Moreover, in the case of al-Qaeda’s violence, alleged religious and secular factors appear to be strongly intertwined and cannot be separated consistently. Thus, the chapter argues, whereas particular beliefs, values and practices deemed ‘religious’ by the perpetrators may fuel jihadist violence in particular contexts, questions of whether religion, as an abstract category, causes or contributes to violence are inconsistent. Instead, it is more fruitful to explore why questions on the role of religion in al-Qaeda’s violence have been so prevalent over the last years. The chapter concludes that, for al-Qaeda, presenting the perceived conflict between Islam and the West as a religious conflict reconfirms and authorises the group’s self-definition and facilitates and legitimises its violence