Analytical sun synchronous low-thrust manoeuvres

Article describes analytical sun synchronous low-thrust manoeuvres

Implementing and evaluating candidate-based invariant generation

The discovery of inductive invariants lies at the heart of static program verification. Presently, many automatic solutions to inductive invariant generation are inflexible, only applicable to certain classes of programs, or unpredictable. An automatic technique that circumvents these deficiencies to some extent is candidate-based invariant generation , whereby a large number of candidate invariants are guessed and then proven to be inductive or rejected using a sound program analyser. This paper describes our efforts to apply candidate-based invariant generation in GPUVerify, a static checker of programs that run on GPUs. We study a set of 383 GPU programs that contain loops, drawn from a number of open source suites and vendor SDKs. Among this set, 253 benchmarks require provision of loop invariants for verification to succeed. We describe the methodology we used to incrementally improve the invariant generation capabilities of GPUVerify to handle these benchmarks, through candidate-based invariant generation , whereby potential program invariants are speculated using cheap static analysis and subsequently either refuted or proven. We also describe a set of experiments that we used to examine the effectiveness of our rules for candidate generation, assessing rules based on their generality (the extent to which they generate candidate invariants), hit rate (the extent to which the generated candidates hold), effectiveness (the extent to which provable candidates actually help in allowing verification to succeed), and influence (the extent to which the success of one generation rule depends on candidates generated by another rule). We believe that our methodology for devising and evaluation candidate generation rules may serve as a useful framework for other researchers interested in candidate-based invariant generation. The candidates produced by GPUVerify help to verify 231 of these 253 programs. An increase in precision, however, has created sluggishness in GPUVerify because more candidates are generated and hence more time is spent on computing those which are inductive invariants. To speed up this process, we have investigated four under-approximating program analyses that aim to reject false candidates quickly and a framework whereby these analyses can run in sequence or in parallel. Across two platforms, running Windows and Linux, our results show that the best combination of these techniques running sequentially speeds up invariant generation across our benchmarks by 1 . 17 × (Windows) and 1 . 01 × (Linux), with per-benchmark best speedups of 93 . 58 × (Windows) and 48 . 34 × (Linux), and worst slowdowns of 10 . 24 × (Windows) and 43 . 31 × (Linux). We find that parallelising the strategies marginally improves overall invariant generation speedups to 1 . 27 × (Windows) and 1 . 11 × (Linux), maintains good best-case speedups of 91 . 18 × (Windows) and 44 . 60 × (Linux), and, importantly, dramatically reduces worst-case slowdowns to 3 . 15 × (Windows) and 3 . 17 × (Linux)

The spin-half Heisenberg antiferromagnet on two Archimedian lattices: From the bounce lattice to the maple-leaf lattice and beyond

We investigate the ground state of the two-dimensional Heisenberg antiferromagnet on two Archimedean lattices, namely, the maple-leaf and bounce lattices as well as a generalized $J$-$J'$ model interpolating between both systems by varying $J'/J$ from $J'/J=0$ (bounce limit) to $J'/J=1$ (maple-leaf limit) and beyond. We use the coupled cluster method to high orders of approximation and also exact diagonalization of finite-sized lattices to discuss the ground-state magnetic long-range order based on data for the ground-state energy, the magnetic order parameter, the spin-spin correlation functions as well as the pitch angle between neighboring spins. Our results indicate that the "pure" bounce ($J'/J=0$) and maple-leaf ($J'/J=1$) Heisenberg antiferromagnets are magnetically ordered, however, with a sublattice magnetization drastically reduced by frustration and quantum fluctuations. We found that magnetic long-range order is present in a wide parameter range $0 \le J'/J \lesssim J'_c/J$ and that the magnetic order parameter varies only weakly with $J'/J$. At $J'_c \approx 1.45 J$ a direct first-order transition to a quantum orthogonal-dimer singlet ground state without magnetic long-range order takes place. The orthogonal-dimer state is the exact ground state in this large-$J'$ regime, and so our model has similarities to the Shastry-Sutherland model. Finally, we use the exact diagonalization to investigate the magnetization curve. We a find a 1/3 magnetization plateau for $J'/J \gtrsim 1.07$ and another one at 2/3 of saturation emerging only at large $J'/J \gtrsim 3$.Comment: 9 pages, 10 figure

Interaction effects and quantum phase transitions in topological insulators

We study strong correlation effects in topological insulators via the Lanczos algorithm, which we utilize to calculate the exact many-particle ground-state wave function and its topological properties. We analyze the simple, noninteracting Haldane model on a honeycomb lattice with known topological properties and demonstrate that these properties are already evident in small clusters. Next, we consider interacting fermions by introducing repulsive nearest-neighbor interactions. A first-order quantum phase transition was discovered at finite interaction strength between the topological band insulator and a topologically trivial Mott insulating phase by use of the fidelity metric and the charge-density-wave structure factor. We construct the phase diagram at $T = 0$ as a function of the interaction strength and the complex phase for the next-nearest-neighbor hoppings. Finally, we consider the Haldane model with interacting hard-core bosons, where no evidence for a topological phase is observed. An important general conclusion of our work is that despite the intrinsic nonlocality of topological phases their key topological properties manifest themselves already in small systems and therefore can be studied numerically via exact diagonalization and observed experimentally, e.g., with trapped ions and cold atoms in optical lattices.Comment: 13 pages, 12 figures. Published versio