Rational Orbits around Charged Black Holes

We show that all eccentric timelike orbits in Reissner-Nordstr\"{o}m spacetime can be classified using a taxonomy that draws upon an isomorphism between periodic orbits and the set of rational numbers. By virtue of the fact that the rationals are dense, the taxonomy can be used to approximate aperiodic orbits with periodic orbits. This may help reduce computational overhead for calculations in gravitational wave astronomy. Our dynamical systems approach enables us to study orbits for both charged and uncharged particles in spite of the fact that charged particle orbits around a charged black hole do not admit a simple one-dimensional effective potential description. Finally, we show that comparing periodic orbits in the RN and Schwarzschild geometries enables us to distinguish charged and uncharged spacetimes by looking only at the orbital dynamics.Comment: 16 pages, 21 figure

PaLM: Scaling Language Modeling with Pathways

Large language models have been shown to achieve remarkable performance across a variety of natural language tasks using few-shot learning, which drastically reduces the number of task-specific training examples needed to adapt the model to a particular application. To further our understanding of the impact of scale on few-shot learning, we trained a 540-billion parameter, densely activated, Transformer language model, which we call Pathways Language Model PaLM. We trained PaLM on 6144 TPU v4 chips using Pathways, a new ML system which enables highly efficient training across multiple TPU Pods. We demonstrate continued benefits of scaling by achieving state-of-the-art few-shot learning results on hundreds of language understanding and generation benchmarks. On a number of these tasks, PaLM 540B achieves breakthrough performance, outperforming the finetuned state-of-the-art on a suite of multi-step reasoning tasks, and outperforming average human performance on the recently released BIG-bench benchmark. A significant number of BIG-bench tasks showed discontinuous improvements from model scale, meaning that performance steeply increased as we scaled to our largest model. PaLM also has strong capabilities in multilingual tasks and source code generation, which we demonstrate on a wide array of benchmarks. We additionally provide a comprehensive analysis on bias and toxicity, and study the extent of training data memorization with respect to model scale. Finally, we discuss the ethical considerations related to large language models and discuss potential mitigation strategies

PaLM 2 Technical Report

We introduce PaLM 2, a new state-of-the-art language model that has better multilingual and reasoning capabilities and is more compute-efficient than its predecessor PaLM. PaLM 2 is a Transformer-based model trained using a mixture of objectives. Through extensive evaluations on English and multilingual language, and reasoning tasks, we demonstrate that PaLM 2 has significantly improved quality on downstream tasks across different model sizes, while simultaneously exhibiting faster and more efficient inference compared to PaLM. This improved efficiency enables broader deployment while also allowing the model to respond faster, for a more natural pace of interaction. PaLM 2 demonstrates robust reasoning capabilities exemplified by large improvements over PaLM on BIG-Bench and other reasoning tasks. PaLM 2 exhibits stable performance on a suite of responsible AI evaluations, and enables inference-time control over toxicity without additional overhead or impact on other capabilities. Overall, PaLM 2 achieves state-of-the-art performance across a diverse set of tasks and capabilities. When discussing the PaLM 2 family, it is important to distinguish between pre-trained models (of various sizes), fine-tuned variants of these models, and the user-facing products that use these models. In particular, user-facing products typically include additional pre- and post-processing steps. Additionally, the underlying models may evolve over time. Therefore, one should not expect the performance of user-facing products to exactly match the results reported in this report

Evidence of market manipulation in the financial crisis

We provide direct evidence of market manipulation at the beginning of the financial crisis in November 2007. The type of manipulation, a "bear raid," would have been prevented by a regulation that was repealed by the Securities and Exchange Commission in July 2007. The regulation, the uptick rule, was designed to prevent manipulation and promote stability and was in force from 1938 as a key part of the government response to the 1929 market crash and its aftermath. On November 1, 2007, Citigroup experienced an unusual increase in trading volume and decrease in price. Our analysis of financial industry data shows that this decline coincided with an anomalous increase in borrowed shares, the selling of which would be a large fraction of the total trading volume. The selling of borrowed shares cannot be explained by news events as there is no corresponding increase in selling by share owners. A similar number of shares were returned on a single day six days later. The magnitude and coincidence of borrowing and returning of shares is evidence of a concerted effort to drive down Citigroup's stock price and achieve a profit, i.e., a bear raid. Interpretations and analyses of financial markets should consider the possibility that the intentional actions of individual actors or coordinated groups can impact market behavior. Markets are not sufficiently transparent to reveal even major market manipulation events. Our results point to the need for regulations that prevent intentional actions that cause markets to deviate from equilibrium and contribute to crashes. Enforcement actions cannot reverse severe damage to the economic system. The current "alternative" uptick rule which is only in effect for stocks dropping by over 10% in a single day is insufficient. Prevention may be achieved through improved availability of market data and the original uptick rule or other transaction limitations.

Solving Quantitative Reasoning Problems with Language Models

Language models have achieved remarkable performance on a wide range of tasks that require natural language understanding. Nevertheless, state-of-the-art models have generally struggled with tasks that require quantitative reasoning, such as solving mathematics, science, and engineering problems at the college level. To help close this gap, we introduce Minerva, a large language model pretrained on general natural language data and further trained on technical content. The model achieves state-of-the-art performance on technical benchmarks without the use of external tools. We also evaluate our model on over two hundred undergraduate-level problems in physics, biology, chemistry, economics, and other sciences that require quantitative reasoning, and find that the model can correctly answer nearly a third of them.Comment: 12 pages, 5 figures + references and appendice

Beyond the imitation game: Quantifying and extrapolating the capabilities of language models

Language models demonstrate both quantitative improvement and new qualitative capabilities with increasing scale. Despite their potentially transformative impact, these new capabilities are as yet poorly characterized. In order to inform future research, prepare for disruptive new model capabilities, and ameliorate socially harmful effects, it is vital that we understand the present and near-future capabilities and limitations of language models. To address this challenge, we introduce the Beyond the Imitation Game benchmark (BIG-bench). BIG-bench currently consists of 204 tasks, contributed by 442 authors across 132 institutions. Task topics are diverse, drawing problems from linguistics, childhood development, math, common-sense reasoning, biology, physics, social bias, software development, and beyond. BIG-bench focuses on tasks that are believed to be beyond the capabilities of current language models. We evaluate the behavior of OpenAI's GPT models, Google-internal dense transformer architectures, and Switch-style sparse transformers on BIG-bench, across model sizes spanning millions to hundreds of billions of parameters. In addition, a team of human expert raters performed all tasks in order to provide a strong baseline. Findings include: model performance and calibration both improve with scale, but are poor in absolute terms (and when compared with rater performance); performance is remarkably similar across model classes, though with benefits from sparsity; tasks that improve gradually and predictably commonly involve a large knowledge or memorization component, whereas tasks that exhibit "breakthrough" behavior at a critical scale often involve multiple steps or components, or brittle metrics; social bias typically increases with scale in settings with ambiguous context, but this can be improved with prompting