Numerical Verification of Affine Systems with up to a Billion Dimensions

Affine systems reachability is the basis of many verification methods. With further computation, methods exist to reason about richer models with inputs, nonlinear differential equations, and hybrid dynamics. As such, the scalability of affine systems verification is a prerequisite to scalable analysis for more complex systems. In this paper, we improve the scalability of affine systems verification, in terms of the number of dimensions (variables) in the system. The reachable states of affine systems can be written in terms of the matrix exponential, and safety checking can be performed at specific time steps with linear programming. Unfortunately, for large systems with many state variables, this direct approach requires an intractable amount of memory while using an intractable amount of computation time. We overcome these challenges by combining several methods that leverage common problem structure. Memory is reduced by exploiting initial states that are not full-dimensional and safety properties (outputs) over a few linear projections of the state variables. Computation time is saved by using numerical simulations to compute only projections of the matrix exponential relevant for the verification problem. Since large systems often have sparse dynamics, we use Krylov-subspace simulation approaches based on the Arnoldi or Lanczos iterations. Our method produces accurate counter-examples when properties are violated and, in the extreme case with sufficient problem structure, can analyze a system with one billion real-valued state variables

From disease incubation to disease receipt:Representing epidemics and race in pre- and post-second world war American cinema (1931–1939 and 1950–1962)

This article analyses continuities and changes in how disease has been instrumentalised in cinema as a way of conceptualizing race—comparing five films depicting epidemics produced before the Second World War and five after. In the 1930s films, non-white populations often passively accept assistance in dealing with epidemic disease—a paternalistic white savior narrative—but not always with “gratitude”, and sometimes direct resistance. Here, epidemics take root in physical sites of economic “underdevelopment”, perpetuated further by perceived “premodern” cultural practices demarcated down the lines of race or ethnicity, and intersect with other gendered and socio-economic categories. After the war, while some cinematic tropes such as the “white knight” continue, other narratives emerge including a shift in emphasis away from the Othered environment as the nexus of disease (the disease’s “incubation”), and towards greater alarm about the appearance of disease within recipient, frequently white, communities

From disease incubation to disease receipt:Representing epidemics and race in pre- and post-second world war American cinema (1931–1939 and 1950–1962)

This article analyses continuities and changes in how disease has been instrumentalised in cinema as a way of conceptualizing race—comparing five films depicting epidemics produced before the Second World War and five after. In the 1930s films, non-white populations often passively accept assistance in dealing with epidemic disease—a paternalistic white savior narrative—but not always with “gratitude”, and sometimes direct resistance. Here, epidemics take root in physical sites of economic “underdevelopment”, perpetuated further by perceived “premodern” cultural practices demarcated down the lines of race or ethnicity, and intersect with other gendered and socio-economic categories. After the war, while some cinematic tropes such as the “white knight” continue, other narratives emerge including a shift in emphasis away from the Othered environment as the nexus of disease (the disease’s “incubation”), and towards greater alarm about the appearance of disease within recipient, frequently white, communities

Staggered Currents in the Vortex Core

We study the electronic structure of the vortex core in the cuprates using the U(1) slave-boson mean-field wavefunctions and their Gutzwiller projection. We conclude that there exists local orbital antiferromagnetic order in the core near optimal doping. We compare the results with that of BCS theory and analyze the spatial dependence of the local tunneling density of states.Comment: 4 pages, 3 figure