Mathematical discoveries from program search with large language models.

Large language models (LLMs) have demonstrated tremendous capabilities in solving complex tasks, from quantitative reasoning to understanding natural language. However, LLMs sometimes suffer from confabulations (or hallucinations), which can result in them making plausible but incorrect statements1,2. This hinders the use of current large models in scientific discovery. Here we introduce FunSearch (short for searching in the function space), an evolutionary procedure based on pairing a pretrained LLM with a systematic evaluator. We demonstrate the effectiveness of this approach to surpass the best-known results in important problems, pushing the boundary of existing LLM-based approaches3. Applying FunSearch to a central problem in extremal combinatorics-the cap set problem-we discover new constructions of large cap sets going beyond the best-known ones, both in finite dimensional and asymptotic cases. This shows that it is possible to make discoveries for established open problems using LLMs. We showcase the generality of FunSearch by applying it to an algorithmic problem, online bin packing, finding new heuristics that improve on widely used baselines. In contrast to most computer search approaches, FunSearch searches for programs that describe how to solve a problem, rather than what the solution is. Beyond being an effective and scalable strategy, discovered programs tend to be more interpretable than raw solutions, enabling feedback loops between domain experts and FunSearch, and the deployment of such programs in real-world applications.

Investigation on Computed Tomography Features of Primary Thymic Atypical Carcinoid Tumors.

OBJECTIVES: The thymic atypical carcinoid tumors (aTCs) are rare. The computed tomography (CT) features of aTC were poorly studied, and researches under this topic with large samples are in a small quantity in the literature. Our aim was to investigate the CT features of aTC. METHODS: Eleven cases of aTC (7 men and 4 women; mean age, 56 ± 16 years) proved by pathology were retrospectively analyzed. All the patients underwent contrast-enhanced CT scans with dual phase (25-30 and 60-70 seconds) after contrast agent injection. Two experienced radiologists analyzed the imaging findings. The observing indexes of CT features were based on standard CT reporting terms of mediastinal masses suspicious for thymic malignancy, particularly on the findings of marginal cystic/necrotic component (MCC) sign and hypervascular sign in tumors. RESULTS: All the tumors were located in the anterior and/or middle mediastinum and heterogeneous parenchyma, with moderate to marked enhancement, and had multiple cystic/necrotic components. The maximum diameter of lesions ranged from 5 to 24 cm. Among the 11 patients involved in this study, 6 patients had distant metastases, all of which were identified as skeletal metastases; 9 cases showed MCC sign only at the margin (3 cases) or at both marginal and central margins (6 cases), and 5 cases showed hypervascular sign. CONCLUSIONS: Our results suggested that the MCC and hypervascular signs should be included as characterization in addition to the standard features for the diagnosis of aTC.

Value of 3.0T MRI T2 mapping combined with SWI for the assessment of early lesions in hemophilic arthropathy.

PURPOSE: To explore the value of magnetic resonance imaging (MRI) T2 mapping combined with susceptibility-weighted imaging (SWI) in detecting early cartilage damage and joint bleeding in the hemophilic arthropathy (HA). METHODS: 147 patients and 56 healthy controls were prospectively recruited. The knees were divided into groups A and B according to the criteria of the International Cartilage Repair Society (ICRS). The Regions of Interest (ROIs) of T2 mapping were drawn for the patella, lateral and medial femoral condyle, and lateral and medial tibial condyle. The T2 values were compared between the patients and control group using one-way ANOVA. The joint count data of International Prophylaxis Study Group (IPSG) scores of conventional and SWI sequences were statistically described using the composition ratio, and the rank sum test was used for the difference analysis. RESULTS: Finally, there were 99 joints in the control group, 135 knees in group A, and 94 knees in group B. There was a significant difference between the T2 value in each subgroup. Comparison of T2 value groups in each cartilage partition, except for group A and group B of the patella, revealed significant differences (all P<0.05). SWI was likely more sensitive than conventional sequences in detecting hemosiderin deposits in hemophilic joints. In addition, the IPSG scores detected by the SWI were generally higher than those of conventional sequences. CONCLUSIONS: MR T2 mapping combined with SWI has great potential to be used for detecting early cartilage damage and micro-hemosiderin deposition in hemophiliac arthropathies and developing preventative treatment plans.

Abnormal brain functional networks in end-stage renal disease patients with cognitive impairment.

INTRODUCTION: Cognitive impairment (CI) is common in patients with end-stage renal disease (ESRD). Neuroimaging studies have demonstrated structural and functional brain alterations underlying CI in patients with ESRD. However, the patterns of change in whole-brain functional networks in ESRD patients with CI remain poorly understood. METHODS: We enrolled 66 patients with ESRD (36 patients with CI and 30 patients without CI) and 48 healthy control subjects (HCs). We calculated the topological properties using a graph theoretical analysis. An analysis of covariance (ANCOVA) was used to compare network metrics among the three groups. Moreover, we analyzed the relationships between altered network measures and clinical variables in ESRD patients with CI. RESULTS: Compared with HCs, both patient groups showed lower local efficiency and small-worldness. ESRD patients had decreased nodal centralities in the default mode regions and right amygdala. Comparison of the two patient groups showed significantly decreased global (small-worldness) and nodal (nodal centralities in the default mode regions) properties in the CI group. Altered nodal centralities in the bilateral medial part of the superior frontal gyrus, left posterior cingulate gyrus, and right precuneus were associated with cognitive performance in the CI group. CONCLUSION: Disrupted brain functional networks were demonstrated in patients with ESRD, which were more severe in those with CI. Moreover, impaired nodal centralities in the default mode regions might underlie CI in patients with ESRD.