Polynomial Norms

In this paper, we study polynomial norms, i.e. norms that are the $d^{\text{th}}$ root of a degree-$d$ homogeneous polynomial $f$. We first show that a necessary and sufficient condition for $f^{1/d}$ to be a norm is for $f$ to be strictly convex, or equivalently, convex and positive definite. Though not all norms come from $d^{\text{th}}$ roots of polynomials, we prove that any norm can be approximated arbitrarily well by a polynomial norm. We then investigate the computational problem of testing whether a form gives a polynomial norm. We show that this problem is strongly NP-hard already when the degree of the form is 4, but can always be answered by testing feasibility of a semidefinite program (of possibly large size). We further study the problem of optimizing over the set of polynomial norms using semidefinite programming. To do this, we introduce the notion of r-sos-convexity and extend a result of Reznick on sum of squares representation of positive definite forms to positive definite biforms. We conclude with some applications of polynomial norms to statistics and dynamical systems

"Snow Scenes'': Exploring the Role of Memory and Place in Commemorating Extreme Winters

Scholars are increasingly focusing on the cultural dimensions of climate, addressing how individuals construct their understanding of climate through local weather. Research often focuses on the importance of widespread conceptualizations of mundane everyday weather, although attention has also been paid to extreme weather events and their potential effect on popular understandings of local climate. This paper introduces the “Snow Scenes” project, which aimed to engage rural communities in Cumbria, England, with their memories of extreme and severe past winter conditions in the region. Collating memories across a wide demographic, using a variety of methods, individual memories were analyzed alongside meteorological and historical records. By exploring these memories and their associated artifacts, this paper aims to better understand the role of memory and place in commemorating extreme winters. First, it is demonstrated how national narratives of exceptional winters are used by individuals as benchmarks against which to gauge conditions. Second, this paper identifies how specific locations and landmarks help to place memories and are shown to be important anchors for individuals’ understanding of their climate. Third, the paper considers how memories of severe winters are often nostalgic in their outlook, with a strong association between snowy winters, childhood, and childhood places. Fourth, it is illustrated how such events are regularly connected to important personal or familial milestones. Finally, the paper reflects on how these local-level experiences of historical extreme events may be central to the shaping of popular understandings of climate and also, by extension, climate change

Partial Recovery in the Graph Alignment Problem

In this paper, we consider the graph alignment problem, which is the problem of recovering, given two graphs, a one-to-one mapping between nodes that maximizes edge overlap. This problem can be viewed as a noisy version of the well-known graph isomorphism problem and appears in many applications, including social network deanonymization and cellular biology. Our focus here is on partial recovery, i.e., we look for a one-to-one mapping which is correct on a fraction of the nodes of the graph rather than on all of them, and we assume that the two input graphs to the problem are correlated Erd\H{o}s-R\'enyi graphs of parameters $(n,q,s)$. Our main contribution is then to give necessary and sufficient conditions on $(n,q,s)$ under which partial recovery is possible with high probability as the number of nodes $n$ goes to infinity. In particular, we show that it is possible to achieve partial recovery in the $nqs=\Theta(1)$ regime under certain additional assumptions. An interesting byproduct of the analysis techniques we develop to obtain the sufficiency result in the partial recovery setting is a tighter analysis of the maximum likelihood estimator for the graph alignment problem, which leads to improved sufficient conditions for exact recovery