FIMO: A Challenge Formal Dataset for Automated Theorem Proving

We present FIMO, an innovative dataset comprising formal mathematical problem statements sourced from the International Mathematical Olympiad (IMO) Shortlisted Problems. Designed to facilitate advanced automated theorem proving at the IMO level, FIMO is currently tailored for the Lean formal language. It comprises 149 formal problem statements, accompanied by both informal problem descriptions and their corresponding LaTeX-based informal proofs. Through initial experiments involving GPT-4, our findings underscore the existing limitations in current methodologies, indicating a substantial journey ahead before achieving satisfactory IMO-level automated theorem proving outcomes

LEGO-Prover: Neural Theorem Proving with Growing Libraries

Despite the success of large language models (LLMs), the task of theorem proving still remains one of the hardest reasoning tasks that is far from being fully solved. Prior methods using language models have demonstrated promising results, but they still struggle to prove even middle school level theorems. One common limitation of these methods is that they assume a fixed theorem library during the whole theorem proving process. However, as we all know, creating new useful theorems or even new theories is not only helpful but crucial and necessary for advancing mathematics and proving harder and deeper results. In this work, we present LEGO-Prover, which employs a growing skill library containing verified lemmas as skills to augment the capability of LLMs used in theorem proving. By constructing the proof modularly, LEGO-Prover enables LLMs to utilize existing skills retrieved from the library and to create new skills during the proving process. These skills are further evolved (by prompting an LLM) to enrich the library on another scale. Modular and reusable skills are constantly added to the library to enable tackling increasingly intricate mathematical problems. Moreover, the learned library further bridges the gap between human proofs and formal proofs by making it easier to impute missing steps. LEGO-Prover advances the state-of-the-art pass rate on miniF2F-valid (48.0% to 57.0%) and miniF2F-test (45.5% to 47.1%). During the proving process, LEGO-Prover also manages to generate over 20,000 skills (theorems/lemmas) and adds them to the growing library. Our ablation study indicates that these newly added skills are indeed helpful for proving theorems, resulting in an improvement from a success rate of 47.1% to 50.4%. We also release our code and all the generated skills

Incorporation of Soft Particles into Lipid Vesicles: Effects of Particle Size and Elasticity

The interaction between particles and lipid biomembranes plays an essential role in many fields such as endocytosis, drug delivery, and intracellular traffic. Here we conduct a theoretical study on the incorporation of elastic particles of different sizes and rigidities into a lipid vesicle through adhesive wrapping. It is shown that while the incorporation of relatively small particles involves smooth shape evolution, the vesicle wrapping of large particles exhibits a discontinuous shape transition, followed by a protrusion of the vesicle membrane at infinitesimal cost of elastic deformation energy. Moreover, softer particles require stronger adhesion energy to achieve successful internalization and delay the onset of discontinuous shape transition to a higher wrapping degree. Depending on the adhesion energy, particle-vesicle size, and rigidity ratios, and the spontaneous curvature of the vesicle, a rich variety of wrapping phase diagrams consisting of stable and metastable states of no-wrapping, partial-wrapping, and full-wrapping are established. The underlying mechanism of the discontinuous shape transformation of the vesicle and the relation between the uptake proneness and uptake efficiency are discussed. These results shed further light on the elasticity effects in cellular uptake of elastic particles and may provide rational design guidelines for controlled endocytosis and diagnostics delivery

Budding of an Adhesive Elastic Particle out of a Lipid Vesicle

The problem of particle budding out of a lipid vesicle is of fundamental importance to our understanding of the biophysical mechanisms involved in viral budding and exocytosis. Here, we present a theoretical study on the outward budding of an adhesive elastic particle out of a lipid vesicle of different spontaneous curvatures. It is shown that a discontinuous shape transformation can occur for budding out of a vesicle with positive spontaneous curvature but not for a vesicle with zero or negative spontaneous curvature, and that softer particles require stronger adhesion energy to achieve full release from the vesicle. Calculations also indicate that the adhesion energy required for full release increases as the spontaneous curvature of the vesicle decreases. A rich variety of budding phase diagrams accounting for the stable or metastable states of no budding, partial budding, and full release are determined. Endocytosis, exocytosis, intracellular budding of elastic particles and related biological implications are discussed. Our results provide physical insights into the biophysical mechanisms of viral budding and exocytosis, and may also provide rational design guidelines for controlled drug delivery systems

Deeply‐Recursive Attention Network for video steganography

Abstract Video steganography plays an important role in secret communication that conceals a secret video in a cover video by perturbing the value of pixels in the cover frames. Imperceptibility is the first and foremost requirement of any steganographic approach. Inspired by the fact that human eyes perceive pixel perturbation differently in different video areas, a novel effective and efficient Deeply‐Recursive Attention Network (DRANet) for video steganography to find suitable areas for information hiding via modelling spatio‐temporal attention is proposed. The DRANet mainly contains two important components, a Non‐Local Self‐Attention (NLSA) block and a Non‐Local Co‐Attention (NLCA) block. Specifically, the NLSA block can select the cover frame areas which are suitable for hiding by computing the correlations among inter‐ and intra‐cover frames. The NLCA block aims to effectively produce the enhanced representations of the secret frames to enhance the robustness of the model and alleviate the influence of different areas in the secret video. Furthermore, the DRANet reduces the model parameters by performing similar operations on the different frames within an input video recursively. Experimental results show the proposed DRANet achieves better performance with fewer parameters than the state‐of‐the‐art competitors

Morphological transformations of vesicles with confined flexible filaments

A fundamental understanding of cell shaping with confined flexible filaments, including microtubules, actin filaments, and engineered nanotubes, has been limited by the complex interplay between the cell membrane and encapsulated filaments. Here, combining theoretical modeling and molecular dynamics simulations, we investigate the packing of an open or closed filament inside a vesicle. Depending on the relative stiffness and size of the filament to the vesicle as well as the osmotic pressure, the vesicle could evolve from an axisymmetric configuration to a general configuration with a maximum of three reflection planes, and the filament could bend in or out of plane or even coil up. A plethora of system morphologies are determined. Morphological phase diagrams predicting conditions of shape and symmetry transitions are established. Organization of actin filaments or bundles, microtubules, and nanotube rings inside vesicles, liposomes, or cells are discussed. Our results provide a theoretical basis to understand cell shaping and cellular stability and to help guide the development and design of artificial cells and biohybrid microrobots.Published versionX.Y. acknowledges the financial support from the National Natural Science Foundation of China (grant nos. 12022207, 11988102, and 12272004), Clinical Medicine Plus X - Young Scholars Project of Peking University, and the Fundamental Research Funds for the Central Universities

Exposure to metals and the disruption of sex hormones in 6–19 years old children: An exploration of mixture effects

Background: Individual metals have been linked to sex hormones disruption, but the associations of metals mixture are rarely examined among children. Methods: A total of 1060 participants of 6–19-year-old who participated in the National Health and Nutrition Examination Survey (2013–2016) were included. Eighteen metals were quantified in the whole blood and urine. Sex hormones were measured in serum, including total testosterone (TT), estradiol (E2), and sex hormone binding globulin (SHBG). In addition, free androgen index (FAI) and the ratio of TT to E2 were calculated. Bayesian kernel machine regression and latent class analysis were performed to assess the associations of metals mixture and exposure patterns of metals at varied levels with sex hormones while adjusting for selected covariates. All analyses were conducted by sex-age and sex-puberty groups to explore the potential sex-dimorphic effects. Results: Exposure to metals mixture was associated with elevated levels of FAI and E2 among 12–19 years old girls. Moreover, the exposure pattern of metals that was characterized by high levels of blood and urinary cadmium, blood manganese, and urinary cobalt was associated with elevated E2 and reduced TT/E2 levels among girls of 12–19 years old. However, the associations of metals mixture with sex hormones were overall nonsignificant among boys. Nevertheless, metals exposure pattern that was characterized by high levels of blood lead, urinary barium, strontium, and lead but comparatively low levels of the other metals was consistently associated with reduced levels of FAI and E2 but elevated levels of TT/E2 and SHBG among boys of 12–19 years old. Conclusion: Metals mixture and exposure patterns that were dominated by high levels of certain metals were associated with sex hormones imbalance among 12–19 years old children in a sex-dimorphic pattern, with the identified individual metals that drove the associations of metals mixture varied by sex

Role of nanoparticle mechanical properties in cancer drug delivery

The physicochemical properties of nanoparticles play critical roles in regulating nano-bio interactions. Whereas the effects of the size, shape, and surface charge of nanoparticles on their biological performances have been extensively investigated, the roles of nanoparticle mechanical properties in drug delivery, which have only been recognized recently, remain the least explored. This review article provides an overview of the impacts of nanoparticle mechanical properties on cancer drug delivery, including (1) basic terminologies of the mechanical properties of nanoparticles and techniques for characterizing these properties; (2) current methods for fabricating nanoparticles with tunable mechanical properties; (3) and studies that highlight key biological performances of stiff and soft nanoparticles, including blood circulation, tumor or tissue targeting, tumor penetration, and cancer cell internalization, with a special emphasis on the underlying mechanisms that control those complicated nano-bio interactions at the cellular, tissue, and organ levels. The interesting research and findings discussed in this review article will offer the research community a better understanding of how this research field evolved during the past years and provide some general guidance on how to design and explore the effects of nanoparticle mechanical properties on nano-bio interactions. These fundamental understandings, will in turn, improve our ability to design better nanoparticles for enhanced drug delivery

Biological and environmental interactions of emerging two-dimensional nanomaterials

Two-dimensional materials have become a major focus in materials chemistry research worldwide with substantial efforts centered on synthesis, property characterization, and technological application. These high-aspect ratio sheet-like solids come in a wide array of chemical compositions, crystal phases, and physical forms, and are anticipated to enable a host of future technologies in areas that include electronics, sensors, coatings, barriers, energy storage and conversion, and biomedicine. A parallel effort has begun to understand the biological and environmental interactions of synthetic nanosheets, both to enable the biomedical developments and to ensure human health and safety for all application fields. This review covers the most recent literature on the biological responses to 2D materials and also draws from older literature on natural lamellar minerals to provide additional insight into the essential chemical behaviors. The article proposes a framework for more systematic investigation of biological behavior in the future, rooted in fundamental materials chemistry and physics. That framework considers three fundamental interaction modes: (i) chemical interactions and phase transformations, (ii) electronic and surface redox interactions, and (iii) physical and mechanical interactions that are unique to near-atomically-thin, high-aspect-ratio solids. Two-dimensional materials are shown to exhibit a wide range of behaviors, which reflect the diversity in their chemical compositions, and many are expected to undergo reactive dissolution processes that will be key to understanding their behaviors and interpreting biological response data. The review concludes with a series of recommendations for high-priority research subtopics at the "bio-nanosheet" interface that we hope will enable safe and successful development of technologies related to two-dimensional nanomaterials