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Motivated by questions from Ehrhart theory, we present new results on discrete equidecomposability. Two rational polygons P and Q are said to be discretely equidecomposable if there exists a piecewise affine-unimodular bijection (equivalently, a piecewise affine-linear bijection that preserves the integer lattice Z2) from P to Q. We develop an invariant for a particular version of this notion called rational finite discrete equidecomposability. We construct triangles that are Ehrhart equivalent but not rationally finitely discretely equidecomposable, thus providing a partial negative answer to a question of Haase–McAllister on whether Ehrhart equivalence implies discrete equidecomposability. Surprisingly, if we delete an edge from each of these triangles, there exists an infinite rational discrete equidecomposability relation between them. Our final section addresses the topic of infinite equidecomposability with concrete examples and a potential setting for further investigation of this phenomenon.

Proving mathematical theorems at the olympiad level represents a notable milestone in human-level automated reasoning 1 – 4 , owing to their reputed difficulty among the world’s best talents in pre-university mathematics. Current machine-learning approaches, however, are not applicable to most mathematical domains owing to the high cost of translating human proofs into machine-verifiable format. The problem is even worse for geometry because of its unique translation challenges 1 , 5 , resulting in severe scarcity of training data. We propose AlphaGeometry, a theorem prover for Euclidean plane geometry that sidesteps the need for human demonstrations by synthesizing millions of theorems and proofs across different levels of complexity. AlphaGeometry is a neuro-symbolic system that uses a neural language model, trained from scratch on our large-scale synthetic data, to guide a symbolic deduction engine through infinite branching points in challenging problems. On a test set of 30 latest olympiad-level problems, AlphaGeometry solves 25, outperforming the previous best method that only solves ten problems and approaching the performance of an average International Mathematical Olympiad (IMO) gold medallist. Notably, AlphaGeometry produces human-readable proofs, solves all geometry problems in the IMO 2000 and 2015 under human expert evaluation and discovers a generalized version of a translated IMO theorem in 2004. A new neuro-symbolic theorem prover for Euclidean plane geometry trained from scratch on millions of synthesized theorems and proofs outperforms the previous best method and reaches the performance of an olympiad gold medallist.

Many real-world applications require artificial agents to compete and coordinate with other agents in complex environments. As a stepping stone to this goal, the domain of StarCraft has emerged as an important challenge for artificial intelligence research, owing to its iconic and enduring status among the most difficult professional esports and its relevance to the real world in terms of its raw complexity and multi-agent challenges. Over the course of a decade and numerous competitions 1 – 3 , the strongest agents have simplified important aspects of the game, utilized superhuman capabilities, or employed hand-crafted sub-systems 4 . Despite these advantages, no previous agent has come close to matching the overall skill of top StarCraft players. We chose to address the challenge of StarCraft using general-purpose learning methods that are in principle applicable to other complex domains: a multi-agent reinforcement learning algorithm that uses data from both human and agent games within a diverse league of continually adapting strategies and counter-strategies, each represented by deep neural networks 5 , 6 . We evaluated our agent, AlphaStar, in the full game of StarCraft II, through a series of online games against human players. AlphaStar was rated at Grandmaster level for all three StarCraft races and above 99.8% of officially ranked human players. AlphaStar uses a multi-agent reinforcement learning algorithm and has reached Grandmaster level, ranking among the top 0.2% of human players for the real-time strategy game StarCraft II.

m6A modification is the most abundant mRNA modifications and plays an integral role in various biological processes in eukaryotes. However, the role of m6A regulators in rheumatoid arthritis remains unknown. To determine the expression of m6A RNA methylation regulators in rheumatoid arthritis and their possible functional and prognostic value. In this study, we performed differential analysis in the comprehensive gene expression database GSE93272 dataset between non-rheumatoid arthritis patients and rheumatoid arthritis patients to obtain 15 important m6A regulators. A random forest model and lasso regression were used to screen the five most important m6A regulators to predict the risk of developing rheumatoid arthritis. After further validation using in vitro qPCR experiments, a nomogram model was developed based on the four most important m6A regulators (ELAVL1, WTAP, YTHDF1, and ALKBH5). Immuno-infiltration analysis and consensus clustering analysis were then performed. An analysis of the decision curve showed that the nomogram model could be beneficial to patients. According to selected important m6A regulators, patients with rheumatoid arthritis were classified into two m6A models (ClusterA and ClusterB) via consensus approach. Activated B cells, CD56dim natural killer cells, immature B cells, monocytes, natural killer T cells, and T lymphocytes were associated with ClusterA in immune infiltration analysis. Importantly, immune infiltration in patients with high ELAVL1 expression was strikingly similar to ClusterA. m6A regulators play a non-negligible role in the development of rheumatoid arthritis. A study of m6A patterns may provide future therapeutic options for rheumatoid arthritis.

This thesis investigates the potential of neural networks as a powerful approach to reasoning. It argues that neural networks can effectively leverage statistical structures to learn useful heuristics and model arbitrary action distributions, making them well-suited for reasoning tasks. To test this claim, the thesis focuses on mathematical reasoning as a well-defined domain with broad applications. The author sets up various benchmarks to define the problem and measure progress in this emerging field. Through extensive experimentation, the thesis demonstrates that neural networks can perform non-trivial mathematical reasoning tasks in both abduction and induction. Additionally, the author shows that various techniques, such as improved inductive bias design, high-level proof sketching, and self-supervised learning, can enhance neural networks' mathematical reasoning capabilities. Overall, this thesis provides compelling evidence that neural networks are a promising tool for mathematical reasoning, with the potential to surpass existing methods in the field.

Stroke, when poor blood flow to the brain results in cell death, is the third leading cause of disability and mortality worldwide, and appears as an unequal distribution in the global population. The cumulative risk of recurrence varies greatly up to 10 years after the first stroke. Carotid atherosclerosis is a major risk factor for stroke. The aim of this study was to investigate and estimate the relationship between carotid atherosclerosis and risk of stroke recurrence in the Chinese population. We performed a systematic review and meta-analysis of randomized controlled trials published from 2000 to 2013, using the following databases: PubMed, Embase, Medline, Wanfang, and the China National Knowledge Infrastructure. The odds ratios with 95% confidence intervals were calculated to examine this strength. A total of 22 studies, including 3,912 patients, 2,506 first-ever cases, and 1,406 recurrent cases, were pooled in this meta-analysis. Our results showed that the frequency of carotid atherosclerosis is higher in recurrent cases than that in the first-ever controls (78.88% vs 59.38%), and the statistical analysis demonstrated significant positive association between carotid atherosclerosis and recurrent cerebral infarction (odds ratio: 2.87; 95% confidence interval: 2.42-3.37; <0.00001) in a fixed-effect model. No significant heterogeneity was observed across all studies. In conclusion, our results showed that carotid atherosclerosis was associated with increased risk of recurrent stroke. However, further well-designed research with large sample sizes is still needed to identify the clear mechanism.

Propositional model counting, or #SAT, is the problem of computing the number of satisfying assignments of a Boolean formula. Many problems from different application areas, including many discrete probabilistic inference problems, can be translated into model counting problems to be solved by #SAT solvers. Exact #SAT solvers, however, are often not scalable to industrial size instances. In this paper, we present Neuro#, an approach for learning branching heuristics to improve the performance of exact #SAT solvers on instances from a given family of problems. We experimentally show that our method reduces the step count on similarly distributed held-out instances and generalizes to much larger instances from the same problem family. It is able to achieve these results on a number of different problem families having very different structures. In addition to step count improvements, Neuro# can also achieve orders of magnitude wall-clock speedups over the vanilla solver on larger instances in some problem families, despite the runtime overhead of querying the model.

Pre-training produces representations that are effective for a wide range of downstream tasks, but it is still unclear what properties of pre-training are necessary for effective gains. Notably, recent work shows that even pre-training on synthetic tasks can achieve significant gains in downstream tasks. In this work, we perform three experiments that iteratively simplify pre-training and show that the simplifications still retain much of its gains. First, building on prior work, we perform a systematic evaluation of three existing synthetic pre-training methods on six downstream tasks. We find the best synthetic pre-training method, LIME, attains an average of \\(67\\%\\) of the benefits of natural pre-training. Second, to our surprise, we find that pre-training on a simple and generic synthetic task defined by the Set function achieves \\(65\\%\\) of the benefits, almost matching LIME. Third, we find that \\(39\\%\\) of the benefits can be attained by using merely the parameter statistics of synthetic pre-training. We release the source code at https://github.com/felixzli/synthetic_pretraining.