Hasil untuk "math-ph"

Dan Hendrycks, Collin Burns, Saurav Kadavath et al.

Many intellectual endeavors require mathematical problem solving, but this skill remains beyond the capabilities of computers. To measure this ability in machine learning models, we introduce MATH, a new dataset of 12,500 challenging competition mathematics problems. Each problem in MATH has a full step-by-step solution which can be used to teach models to generate answer derivations and explanations. To facilitate future research and increase accuracy on MATH, we also contribute a large auxiliary pretraining dataset which helps teach models the fundamentals of mathematics. Even though we are able to increase accuracy on MATH, our results show that accuracy remains relatively low, even with enormous Transformer models. Moreover, we find that simply increasing budgets and model parameter counts will be impractical for achieving strong mathematical reasoning if scaling trends continue. While scaling Transformers is automatically solving most other text-based tasks, scaling is not currently solving MATH. To have more traction on mathematical problem solving we will likely need new algorithmic advancements from the broader research community.

Arkil Patel, S. Bhattamishra, Navin Goyal

The problem of designing NLP solvers for math word problems (MWP) has seen sustained research activity and steady gains in the test accuracy. Since existing solvers achieve high performance on the benchmark datasets for elementary level MWPs containing one-unknown arithmetic word problems, such problems are often considered “solved” with the bulk of research attention moving to more complex MWPs. In this paper, we restrict our attention to English MWPs taught in grades four and lower. We provide strong evidence that the existing MWP solvers rely on shallow heuristics to achieve high performance on the benchmark datasets. To this end, we show that MWP solvers that do not have access to the question asked in the MWP can still solve a large fraction of MWPs. Similarly, models that treat MWPs as bag-of-words can also achieve surprisingly high accuracy. Further, we introduce a challenge dataset, SVAMP, created by applying carefully chosen variations over examples sampled from existing datasets. The best accuracy achieved by state-of-the-art models is substantially lower on SVAMP, thus showing that much remains to be done even for the simplest of the MWPs.

An Yang, Beichen Zhang, Binyuan Hui et al.

In this report, we present a series of math-specific large language models: Qwen2.5-Math and Qwen2.5-Math-Instruct-1.5B/7B/72B. The core innovation of the Qwen2.5 series lies in integrating the philosophy of self-improvement throughout the entire pipeline, from pre-training and post-training to inference: (1) During the pre-training phase, Qwen2-Math-Instruct is utilized to generate large-scale, high-quality mathematical data. (2) In the post-training phase, we develop a reward model (RM) by conducting massive sampling from Qwen2-Math-Instruct. This RM is then applied to the iterative evolution of data in supervised fine-tuning (SFT). With a stronger SFT model, it's possible to iteratively train and update the RM, which in turn guides the next round of SFT data iteration. On the final SFT model, we employ the ultimate RM for reinforcement learning, resulting in the Qwen2.5-Math-Instruct. (3) Furthermore, during the inference stage, the RM is used to guide sampling, optimizing the model's performance. Qwen2.5-Math-Instruct supports both Chinese and English, and possess advanced mathematical reasoning capabilities, including Chain-of-Thought (CoT) and Tool-Integrated Reasoning (TIR). We evaluate our models on 10 mathematics datasets in both English and Chinese, such as GSM8K, MATH, GaoKao, AMC23, and AIME24, covering a range of difficulties from grade school level to math competition problems.

Peiyi Wang, Lei Li, Zhihong Shao et al.

In this paper, we present an innovative process-oriented math process reward model called \textbf{Math-Shepherd}, which assigns a reward score to each step of math problem solutions. The training of Math-Shepherd is achieved using automatically constructed process-wise supervision data, breaking the bottleneck of heavy reliance on manual annotation in existing work. We explore the effectiveness of Math-Shepherd in two scenarios: 1) \textit{Verification}: Math-Shepherd is utilized for reranking multiple outputs generated by Large Language Models (LLMs); 2) \textit{Reinforcement Learning}: Math-Shepherd is employed to reinforce LLMs with step-by-step Proximal Policy Optimization (PPO). With Math-Shepherd, a series of open-source LLMs demonstrates exceptional performance. For instance, the step-by-step PPO with Math-Shepherd significantly improves the accuracy of Mistral-7B (77.9\%$\to$84.1\% on GSM8K and 28.6\%$\to$33.0\% on MATH). The accuracy can be further enhanced to 89.1\% and 43.5\% on GSM8K and MATH with the verification of Math-Shepherd, respectively. We believe that automatic process supervision holds significant potential for the future evolution of LLMs.

Renrui Zhang, Dongzhi Jiang, Yichi Zhang et al.

The remarkable progress of Multi-modal Large Language Models (MLLMs) has garnered unparalleled attention, due to their superior performance in visual contexts. However, their capabilities in visual math problem-solving remain insufficiently evaluated and understood. We investigate current benchmarks to incorporate excessive visual content within textual questions, which potentially assist MLLMs in deducing answers without truly interpreting the input diagrams. To this end, we introduce MathVerse, an all-around visual math benchmark designed for an equitable and in-depth evaluation of MLLMs. We meticulously collect 2,612 high-quality, multi-subject math problems with diagrams from publicly available sources. Each problem is then transformed by human annotators into six distinct versions, each offering varying degrees of information content in multi-modality, contributing to 15K test samples in total. This approach allows MathVerse to comprehensively assess whether and how much MLLMs can truly understand the visual diagrams for mathematical reasoning. In addition, we propose a Chain-of-Thought (CoT) evaluation strategy for a fine-grained assessment of the output answers. Rather than naively judging True or False, we employ GPT-4(V) to adaptively extract crucial reasoning steps, and then score each step with detailed error analysis, which can reveal the intermediate CoT reasoning quality by MLLMs. We hope the MathVerse benchmark may provide unique insights to guide the future development of MLLMs. Project page: https://mathverse-cuhk.github.io

Ke Wang, Junting Pan, Weikang Shi et al.

Recent advancements in Large Multimodal Models (LMMs) have shown promising results in mathematical reasoning within visual contexts, with models approaching human-level performance on existing benchmarks such as MathVista. However, we observe significant limitations in the diversity of questions and breadth of subjects covered by these benchmarks. To address this issue, we present the MATH-Vision (MATH-V) dataset, a meticulously curated collection of 3,040 high-quality mathematical problems with visual contexts sourced from real math competitions. Spanning 16 distinct mathematical disciplines and graded across 5 levels of difficulty, our dataset provides a comprehensive and diverse set of challenges for evaluating the mathematical reasoning abilities of LMMs. Through extensive experimentation, we unveil a notable performance gap between current LMMs and human performance on MATH-V, underscoring the imperative for further advancements in LMMs. Moreover, our detailed categorization allows for a thorough error analysis of LMMs, offering valuable insights to guide future research and development. The project is available at https://mathvision-cuhk.github.io

Xiang Yue, Xingwei Qu, Ge Zhang et al.

We introduce MAmmoTH, a series of open-source large language models (LLMs) specifically tailored for general math problem-solving. The MAmmoTH models are trained on MathInstruct, our meticulously curated instruction tuning dataset. MathInstruct is compiled from 13 math datasets with intermediate rationales, six of which have rationales newly curated by us. It presents a unique hybrid of chain-of-thought (CoT) and program-of-thought (PoT) rationales, and also ensures extensive coverage of diverse fields in math. The hybrid of CoT and PoT not only unleashes the potential of tool use but also allows different thought processes for different math problems. As a result, the MAmmoTH series substantially outperform existing open-source models on nine mathematical reasoning datasets across all scales with an average accuracy gain between 16% and 32%. Remarkably, our MAmmoTH-7B model reaches 33% on MATH (a competition-level dataset), which exceeds the best open-source 7B model (WizardMath) by 23%, and the MAmmoTH-34B model achieves 44% accuracy on MATH, even surpassing GPT-4's CoT result. Our work underscores the importance of diverse problem coverage and the use of hybrid rationales in developing superior math generalist models.

Xinyu Guan, L. Zhang, Yifei Liu et al.

We present rStar-Math to demonstrate that small language models (SLMs) can rival or even surpass the math reasoning capability of OpenAI o1, without distillation from superior models. rStar-Math achieves this by exercising"deep thinking"through Monte Carlo Tree Search (MCTS), where a math policy SLM performs test-time search guided by an SLM-based process reward model. rStar-Math introduces three innovations to tackle the challenges in training the two SLMs: (1) a novel code-augmented CoT data sythesis method, which performs extensive MCTS rollouts to generate step-by-step verified reasoning trajectories used to train the policy SLM; (2) a novel process reward model training method that avoids na\"ive step-level score annotation, yielding a more effective process preference model (PPM); (3) a self-evolution recipe in which the policy SLM and PPM are built from scratch and iteratively evolved to improve reasoning capabilities. Through 4 rounds of self-evolution with millions of synthesized solutions for 747k math problems, rStar-Math boosts SLMs' math reasoning to state-of-the-art levels. On the MATH benchmark, it improves Qwen2.5-Math-7B from 58.8% to 90.0% and Phi3-mini-3.8B from 41.4% to 86.4%, surpassing o1-preview by +4.5% and +0.9%. On the USA Math Olympiad (AIME), rStar-Math solves an average of 53.3% (8/15) of problems, ranking among the top 20% the brightest high school math students. Code and data will be available at https://github.com/microsoft/rStar.

Runqi Qiao, Qiuna Tan, Guanting Dong et al.

Visual mathematical reasoning, as a fundamental visual reasoning ability, has received widespread attention from the Large Multimodal Models (LMMs) community. Existing benchmarks, such as MathVista and MathVerse, focus more on the result-oriented performance but neglect the underlying principles in knowledge acquisition and generalization. Inspired by human-like mathematical reasoning, we introduce WE-MATH, the first benchmark specifically designed to explore the problem-solving principles beyond end-to-end performance. We meticulously collect and categorize 6.5K visual math problems, spanning 67 hierarchical knowledge concepts and five layers of knowledge granularity. We decompose composite problems into sub-problems according to the required knowledge concepts and introduce a novel four-dimensional metric, namely Insufficient Knowledge (IK), Inadequate Generalization (IG), Complete Mastery (CM), and Rote Memorization (RM), to hierarchically assess inherent issues in LMMs' reasoning process. With WE-MATH, we conduct a thorough evaluation of existing LMMs in visual mathematical reasoning and reveal a negative correlation between solving steps and problem-specific performance. We confirm the IK issue of LMMs can be effectively improved via knowledge augmentation strategies. More notably, the primary challenge of GPT-4o has significantly transitioned from IK to IG, establishing it as the first LMM advancing towards the knowledge generalization stage. In contrast, other LMMs exhibit a marked inclination towards Rote Memorization - they correctly solve composite problems involving multiple knowledge concepts yet fail to answer sub-problems. We anticipate that WE-MATH will open new pathways for advancements in visual mathematical reasoning for LMMs. The WE-MATH data and evaluation code are available at https://github.com/We-Math/We-Math.

Bofei Gao, Feifan Song, Zhe Yang et al.

Recent advancements in large language models (LLMs) have led to significant breakthroughs in mathematical reasoning capabilities. However, existing benchmarks like GSM8K or MATH are now being solved with high accuracy (e.g., OpenAI o1 achieves 94.8\% on MATH dataset), indicating their inadequacy for truly challenging these models. To bridge this gap, we propose a comprehensive and challenging benchmark specifically designed to assess LLMs' mathematical reasoning at the Olympiad level. Unlike existing Olympiad-related benchmarks, our dataset focuses exclusively on mathematics and comprises a vast collection of 4428 competition-level problems with rigorous human annotation. These problems are meticulously categorized into over 33 sub-domains and span more than 10 distinct difficulty levels, enabling a holistic assessment of model performance in Olympiad-mathematical reasoning. Furthermore, we conducted an in-depth analysis based on this benchmark. Our experimental results show that even the most advanced models, OpenAI o1-mini and OpenAI o1-preview, struggle with highly challenging Olympiad-level problems, with 60.54\% and 52.55\% accuracy, highlighting significant challenges in Olympiad-level mathematical reasoning.

Shubham Toshniwal, Wei Du, Ivan Moshkov et al.

Mathematical reasoning continues to be a critical challenge in large language model (LLM) development with significant interest. However, most of the cutting-edge progress in mathematical reasoning with LLMs has become \emph{closed-source} due to lack of access to training data. This lack of data access limits researchers from understanding the impact of different choices for synthesizing and utilizing the data. With the goal of creating a high-quality finetuning (SFT) dataset for math reasoning, we conduct careful ablation experiments on data synthesis using the recently released \texttt{Llama3.1} family of models. Our experiments show that: (a) solution format matters, with excessively verbose solutions proving detrimental to SFT performance, (b) data generated by a strong teacher outperforms equally-sized data generated by a weak student model, (c) SFT is robust to low-quality solutions, allowing for imprecise data filtering, and (d) question diversity is crucial for achieving data scaling gains. Based on these insights, we create the OpenMathInstruct-2 dataset, which consists of 14M question-solution pairs ($\approx$ 600K unique questions), making it nearly eight times larger than the previous largest open-source math reasoning dataset. Finetuning the \texttt{Llama-3.1-8B-Base} using OpenMathInstruct-2 outperforms \texttt{Llama3.1-8B-Instruct} on MATH by an absolute 15.9\% (51.9\% $\rightarrow$ 67.8\%). Finally, to accelerate the open-source efforts, we release the code, the finetuned models, and the OpenMathInstruct-2 dataset under a commercially permissive license.

Shubham Toshniwal, Ivan Moshkov, Sean Narenthiran et al.

Recent work has shown the immense potential of synthetically generated datasets for training large language models (LLMs), especially for acquiring targeted skills. Current large-scale math instruction tuning datasets such as MetaMathQA (Yu et al., 2024) and MAmmoTH (Yue et al., 2024) are constructed using outputs from closed-source LLMs with commercially restrictive licenses. A key reason limiting the use of open-source LLMs in these data generation pipelines has been the wide gap between the mathematical skills of the best closed-source LLMs, such as GPT-4, and the best open-source LLMs. Building on the recent progress in open-source LLMs, our proposed prompting novelty, and some brute-force scaling, we construct OpenMathInstruct-1, a math instruction tuning dataset with 1.8M problem-solution pairs. The dataset is constructed by synthesizing code-interpreter solutions for GSM8K and MATH, two popular math reasoning benchmarks, using the recently released and permissively licensed Mixtral model. Our best model, OpenMath-CodeLlama-70B, trained on a subset of OpenMathInstruct-1, achieves a score of 84.6% on GSM8K and 50.7% on MATH, which is competitive with the best gpt-distilled models. We release our code, models, and the OpenMathInstruct-1 dataset under a commercially permissive license.

Wenhao Shi, Zhiqiang Hu, Yi Bin et al.

Large language models (LLMs) have demonstrated impressive reasoning capabilities, particularly in textual mathematical problem-solving. However, existing open-source image instruction fine-tuning datasets, containing limited question-answer pairs per image, do not fully exploit visual information to enhance the multimodal mathematical reasoning capabilities of Multimodal LLMs (MLLMs). To bridge this gap, we address the lack of high-quality, diverse multimodal mathematical datasets by collecting 40K high-quality images with question-answer pairs from 24 existing datasets and synthesizing 320K new pairs, creating the MathV360K dataset, which enhances both the breadth and depth of multimodal mathematical questions. We introduce Math-LLaVA, a LLaVA-1.5-based model fine-tuned with MathV360K. This novel approach significantly improves the multimodal mathematical reasoning capabilities of LLaVA-1.5, achieving a 19-point increase and comparable performance to GPT-4V on MathVista's minitest split, and yielding leading performance on Math-V and MathVerse. Furthermore, Math-LLaVA demonstrates enhanced generalizability, showing substantial improvements on the MMMU benchmark. Our research highlights the importance of dataset diversity and synthesis in advancing MLLMs' mathematical reasoning abilities. The code and data are available at: \url{https://github.com/HZQ950419/Math-LLaVA}.

Huaiyuan Ying, Shuo Zhang, Linyang Li et al.

The math abilities of large language models can represent their abstract reasoning ability. In this paper, we introduce and open-source our math reasoning LLMs InternLM-Math which is continue pre-trained from InternLM2. We unify chain-of-thought reasoning, reward modeling, formal reasoning, data augmentation, and code interpreter in a unified seq2seq format and supervise our model to be a versatile math reasoner, verifier, prover, and augmenter. These abilities can be used to develop the next math LLMs or self-iteration. InternLM-Math obtains open-sourced state-of-the-art performance under the setting of in-context learning, supervised fine-tuning, and code-assisted reasoning in various informal and formal benchmarks including GSM8K, MATH, Hungary math exam, MathBench-ZH, and MiniF2F. Our pre-trained model achieves 30.3 on the MiniF2F test set without fine-tuning. We further explore how to use LEAN to solve math problems and study its performance under the setting of multi-task learning which shows the possibility of using LEAN as a unified platform for solving and proving in math. Our models, codes, and data are released at \url{https://github.com/InternLM/InternLM-Math}.

Arindam Mitra, Hamed Khanpour, Corby Rosset et al.

Mathematical word problem-solving has long been recognized as a complex task for small language models (SLMs). A recent study hypothesized that the smallest model size, needed to achieve over 80% accuracy on the GSM8K benchmark, is 34 billion parameters. To reach this level of performance with smaller models, researcher often train SLMs to generate Python code or use tools to help avoid calculation errors. Additionally, they employ ensembling, where outputs of up to 100 model runs are combined to arrive at a more accurate result. Result selection is done using consensus, majority vote or a separate a verifier model used in conjunction with the SLM. Ensembling provides a substantial boost in accuracy but at a significant cost increase with multiple calls to the model (e.g., Phi-GSM uses top-48 to boost the performance from 68.2 to 81.5). In this work, we present Orca-Math, a 7-billion-parameter SLM based on the Mistral-7B, which achieves 86.81% on GSM8k without the need for multiple model calls or the use of verifiers, code execution or any other external tools. Our approach has the following key elements: (1) A high quality synthetic dataset of 200K math problems created using a multi-agent setup where agents collaborate to create the data, (2) An iterative learning techniques that enables the SLM to practice solving problems, receive feedback on its solutions and learn from preference pairs incorporating the SLM solutions and the feedback. When trained with Supervised Fine-Tuning alone, Orca-Math achieves 81.50% on GSM8k pass@1 metric. With iterative preference learning, Orca-Math achieves 86.81% pass@1. Orca-Math surpasses the performance of significantly larger models such as LLAMA-2-70B, WizardMath-70B, Gemini-Pro, ChatGPT-3.5. It also significantly outperforms other smaller models while using much smaller data (hundreds of thousands vs. millions of problems).

Chen Li, Weiqi Wang, Jingcheng Hu et al.

Mathematical capabilities were previously believed to emerge in common language models only at a very large scale or require extensive math-related pre-training. This paper shows that the LLaMA-2 7B model with common pre-training already exhibits strong mathematical abilities, as evidenced by its impressive accuracy of 97.7% and 72.0% on the GSM8K and MATH benchmarks, respectively, when selecting the best response from 256 random generations. The primary issue with the current base model is the difficulty in consistently eliciting its inherent mathematical capabilities. Notably, the accuracy for the first answer drops to 49.5% and 7.9% on the GSM8K and MATH benchmarks, respectively. We find that simply scaling up the SFT data can significantly enhance the reliability of generating correct answers. However, the potential for extensive scaling is constrained by the scarcity of publicly available math questions. To overcome this limitation, we employ synthetic data, which proves to be nearly as effective as real data and shows no clear saturation when scaled up to approximately one million samples. This straightforward approach achieves an accuracy of 82.6% on GSM8K and 40.6% on MATH using LLaMA-2 7B models, surpassing previous models by 14.2% and 20.8%, respectively. We also provide insights into scaling behaviors across different reasoning complexities and error types.

Kaixuan Huang, Jiacheng Guo, Zihao Li et al.

Large language models have demonstrated impressive performance on challenging mathematical reasoning tasks, which has triggered the discussion of whether the performance is achieved by true reasoning capability or memorization. To investigate this question, prior work has constructed mathematical benchmarks when questions undergo simple perturbations -- modifications that still preserve the underlying reasoning patterns of the solutions. However, no work has explored hard perturbations, which fundamentally change the nature of the problem so that the original solution steps do not apply. To bridge the gap, we construct MATH-P-Simple and MATH-P-Hard via simple perturbation and hard perturbation, respectively. Each consists of 279 perturbed math problems derived from level-5 (hardest) problems in the MATH dataset (Hendrycksmath et. al., 2021). We observe significant performance drops on MATH-P-Hard across various models, including o1-mini (-16.49%) and gemini-2.0-flash-thinking (-12.9%). We also raise concerns about a novel form of memorization where models blindly apply learned problem-solving skills without assessing their applicability to modified contexts. This issue is amplified when using original problems for in-context learning. We call for research efforts to address this challenge, which is critical for developing more robust and reliable reasoning models.

María Zúñiga-Navarro, Somaris E. Quintana, Luis A. García-Zapateiro

The preparation and characterization of phenolic extracts from <i>Averrhoa carambola</i> were performed to develop enriched mayonnaise-type emulsions, evaluating the effect on their physicochemical, rheological, and microstructural properties. Extracts were obtained by Ultrasound-Assisted Extraction (UAE) employing different ethanol:water ratios, followed by the analysis of their Total Phenolic Content (TPC) and Antioxidant Activity (AA). The 50:50 extract (AEt50) exhibiting the highest bioactivity was selected for the development of enriched mayonnaise, which was then subjected to stability, physicochemical, rheological, and microstructural analyses. Extraction yields ranged from 13% to 28%, with TPC values spanning 3251 to 4661 mg GAE/g of extract, and AA values between 49.25 and 81.67 µMol Trolox/g of extract. Subsequently, the strategic incorporation of the extract, coupled with pH adjustment, successfully maintained the pH of the final products at approximately 4.63 and preserved emulsion stability. This process resulted in a significant, dose-dependent increase in TPC and AA in the mayonnaise, with the highest concentration achieving nearly 9.0 mg GAE/g and the antioxidant activity de 60.0 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="sans-serif">μ</mi></mrow></semantics></math></inline-formula>Mol Trolox/g. The microstructural integrity was maintained, with all droplet sizes remaining under 4 µm, though a visible change in color (ΔE) was observed. All samples exhibited shear-thinning, non-Newtonian behavior, accurately fitted to the Ostwald–de Waele model (R² > 0.982), and demonstrated a dominant elastic structure (G′ > G″) characteristic of high-quality solid-like gels. Thus, the incorporation of <i>Averrhoa carambola</i> extracts is a technically viable and effective alternative to develop stable food products enriched with functional bioactive compounds.

P. Le Pape, B. Baptiste, G. Radtke et al.

<p>In surface soils and sediments, iron monosulfide (FeS) species, including nanocrystalline mackinawite, tend to quickly form in the presence of iron and sulfide in anoxic conditions. As such, FeS species are the main precursors for the formation of other iron sulfides such as Fe<span class="inline-formula">₃</span>S<span class="inline-formula">₄</span> greigite and FeS<span class="inline-formula">₂</span> pyrite, which are ubiquitous in surface sedimentary environments. It is known that, under prolonged aging under reducing conditions in a sulfidic aqueous medium, FeS species can evolve into crystalline mackinawite. However, the possible influence of pH on the evolution of mackinawite under such anoxic low-temperature conditions relevant to sedimentary (sub)surface environments has not been investigated yet. In this study, we used Rietveld refinement and pair distribution function analysis (PDF) of synchrotron-based X-ray powder diffraction (XRD) patterns to derive the mean coherent domain (MCD) size of mackinawite after aging under various pH conditions and X-ray absorption near-edge structure (XANES) spectroscopy at the S and Fe <span class="inline-formula"><i>K</i></span>-edges to study the structural and electronic properties. Moreover, in order to strengthen our interpretations, we confirmed the shape and relative energy of pre-edge features in the S <span class="inline-formula"><i>K</i></span>-edge XANES spectra of mackinawite (FeS) and pyrite (FeS<span class="inline-formula">₂</span>) model compounds via first-principle calculations. Our results show that, after FeS has precipitated from aqueous Fe(II) and <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M7" display="inline" overflow="scroll" dspmath="mathml"><mrow class="chem"><msub><mi mathvariant="normal">H</mi><mn mathvariant="normal">2</mn></msub><mi mathvariant="normal">S</mi><mo>/</mo><msup><mi mathvariant="normal">HS</mi><mo>-</mo></msup></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="50pt" height="14pt" class="svg-formula" dspmath="mathimg" md5hash="4c2e61bebf3aa9ce1218ba765298f1ba"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ejm-38-135-2026-ie00001.svg" width="50pt" height="14pt" src="ejm-38-135-2026-ie00001.png"/></svg:svg></span></span> in a saline medium at pH 7.1, aqueous aging at the same pH over 47 d results in the formation of nanocrystalline mackinawite (MCD<span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M8" display="inline" overflow="scroll" dspmath="mathml"><mrow><msub><mi/><mrow><mi>a</mi><mi>b</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">11.5</mn><mo>±</mo><mn mathvariant="normal">0.1</mn></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="73pt" height="12pt" class="svg-formula" dspmath="mathimg" md5hash="2118cfa36d0a1e9457e8bd91777ce44c"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ejm-38-135-2026-ie00002.svg" width="73pt" height="12pt" src="ejm-38-135-2026-ie00002.png"/></svg:svg></span></span> nm; MCD<span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M9" display="inline" overflow="scroll" dspmath="mathml"><mrow><msub><mi/><mi>c</mi></msub><mo>=</mo><mn mathvariant="normal">7.1</mn><mo>±</mo><mn mathvariant="normal">0.1</mn></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="62pt" height="12pt" class="svg-formula" dspmath="mathimg" md5hash="ac2221155b0d94eb38546465fb7c0007"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ejm-38-135-2026-ie00003.svg" width="62pt" height="12pt" src="ejm-38-135-2026-ie00003.png"/></svg:svg></span></span> nm). When Na<span class="inline-formula">₂</span>S is added into the solution to reach pH 9.7 after FeS has precipitated at pH 7.1, no other Fe sulfide is observed during the aging phase, and mackinawite particles are of smaller size (MCD<span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M11" display="inline" overflow="scroll" dspmath="mathml"><mrow><msub><mi/><mrow><mi>a</mi><mi>b</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">7.9</mn><mo>±</mo><mn mathvariant="normal">0.1</mn></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="67pt" height="12pt" class="svg-formula" dspmath="mathimg" md5hash="fa999fd907eb689fcbdb8c12680b6237"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ejm-38-135-2026-ie00004.svg" width="67pt" height="12pt" src="ejm-38-135-2026-ie00004.png"/></svg:svg></span></span> nm; MCD<span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M12" display="inline" overflow="scroll" dspmath="mathml"><mrow><msub><mi/><mi>c</mi></msub><mo>=</mo><mn mathvariant="normal">4.6</mn><mo>±</mo><mn mathvariant="normal">0.1</mn></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="62pt" height="12pt" class="svg-formula" dspmath="mathimg" md5hash="31f119bd6c4630f13e88f111abe3dff8"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ejm-38-135-2026-ie00005.svg" width="62pt" height="12pt" src="ejm-38-135-2026-ie00005.png"/></svg:svg></span></span> nm). In this sample, an additional weak and broad peak appears at <span class="inline-formula"><i>d</i>=10.5</span> Å that could be interpreted as being due to either lattice expansion at the particle boundaries or a double-cell super-structure. When H<span class="inline-formula">⁺</span> is added as HCl to reach pH 5.1 before the aging phase, the size of mackinawite particles increases (MCD<span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M15" display="inline" overflow="scroll" dspmath="mathml"><mrow><msub><mi/><mrow><mi>a</mi><mi>b</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">13.0</mn><mo>±</mo><mn mathvariant="normal">0.2</mn></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="73pt" height="12pt" class="svg-formula" dspmath="mathimg" md5hash="8aadd77d956dbcf7a2e1acff85e27c33"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ejm-38-135-2026-ie00006.svg" width="73pt" height="12pt" src="ejm-38-135-2026-ie00006.png"/></svg:svg></span></span> nm; MCD<span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M16" display="inline" overflow="scroll" dspmath="mathml"><mrow><msub><mi/><mi>c</mi></msub><mo>=</mo><mn mathvariant="normal">8.1</mn><mo>±</mo><mn mathvariant="normal">0.2</mn></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="62pt" height="12pt" class="svg-formula" dspmath="mathimg" md5hash="9e74d8dc994f8d90fbcdf8b67208ec4c"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="ejm-38-135-2026-ie00007.svg" width="62pt" height="12pt" src="ejm-38-135-2026-ie00007.png"/></svg:svg></span></span> nm), and a fraction transforms into greigite (Fe<span class="inline-formula">₃</span>S<span class="inline-formula">₄</span>). This reaction is accompanied by a pH increase to 6.4, likely because of H<span class="inline-formula">⁺</span> consumption, which suggests that Fe(II) in FeS would serve as an electron donor and that H<span class="inline-formula">⁺</span> would serve as an electron acceptor. The calculated electronic structure of mackinawite shows partly filled Fe-3<span class="inline-formula"><i>d</i></span> states, which supports the fact that acidic aging conditions are favorable for Fe(II) to act as an electron donor. We propose and further discuss the fact that the formation of greigite from nanocrystalline mackinawite could result in H<span class="inline-formula">₂</span> production as, for instance, observed for anoxic corrosion of zero-valent Fe at higher temperatures. Greigite has been designated in the literature either as an intermediate towards pyrite formation or as a mineralogical endmember in another reaction route. Our observations raise the question of the existence of such a reaction producing Fe<span class="inline-formula">₃</span>S<span class="inline-formula">₄</span> and H<span class="inline-formula">₂</span> in reducing sedimentary (micro)environments across geological times. In addition, the metallic character of mackinawite suggests that Fe(II) oxidation to Fe(III) by H<span class="inline-formula">⁺</span> in this mineral species could proceed without the need for another oxidizing agent. Although the possible formation of pyrite from greigite<span id="page136"/> would require further studies on extended aging time and/or under more acid-sulfidic conditions, our findings could have implications for the understanding of the initial steps of the H<span class="inline-formula">₂</span>S pathway to pyrite.</p>

Yang Liu, Zhenzhen He, Jing Fan et al.

In this study, we extract a 7-day binned <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula>-ray light curve from 2008 August to 2019 March in the energy range 0.1–300 GeV and identify four outburst periods with peak flux of >8.0<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mspace width="3.33333pt"></mspace><mo>×</mo><mspace width="3.33333pt"></mspace><msup><mn>10</mn><mrow><mo>−</mo><mn>7</mn></mrow></msup></mrow></semantics></math></inline-formula> ph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>cm</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mtext> </mtext><msup><mi mathvariant="normal">s</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></semantics></math></inline-formula>. Four active states in the optical are also marked during this period. The fastest variability timescale suggests the emission region radius is <i>R</i> ∼ 2.4<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mspace width="3.33333pt"></mspace><mo>×</mo><mspace width="3.33333pt"></mspace><msup><mn>10</mn><mn>16</mn></msup></mrow></semantics></math></inline-formula> cm, and the observed emission region lies within <0.7 pc distance from the central engine. The majority of short-timescale flares exhibit a symmetric temporal profile, implying that the rise and decay timescales are dominated by disturbances caused by dense plasma blobs passing through the standing shock front in the jet region. To understand the properties of the source jets, we employ a standard one-zone leptonic scenario to model the broadband spectral energy distributions (SEDs) of flaring periods and determine that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula>-ray spectrum is better reproduced when the dissipation region of the jet is located within the molecular torus (MT). The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula>-ray spectra from the outburst phases show an obvious spectral break with a break energy between 3.00 and 7.08 GeV, which may be attributed to an intrinsic break in the energy distribution of radiating particles. The studies of the survival time of a sheet before being destroyed by the turbulent motions of plasma (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>τ</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub><mo>∼</mo><mn>2.9</mn><mo>×</mo><msup><mn>10</mn><mn>4</mn></msup></mrow></semantics></math></inline-formula> s), the shock acceleration time (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>t</mi><mrow><mi>a</mi><mi>c</mi><mi>c</mi></mrow></msub><mo>∼</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>4.3</mn><mo>×</mo><mspace width="3.33333pt"></mspace><msup><mn>10</mn><mn>4</mn></msup></mrow></semantics></math></inline-formula> s), and the minimum interaction height (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Z</mi><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub></semantics></math></inline-formula> ≈ 2.57–4.55<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mspace width="3.33333pt"></mspace><mo>×</mo><mspace width="3.33333pt"></mspace><msup><mn>10</mn><mn>17</mn></msup></mrow></semantics></math></inline-formula> cm > <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><mrow><mi>B</mi><mi>L</mi><mi>R</mi></mrow></msub></semantics></math></inline-formula> ∼ 1.0<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mspace width="3.33333pt"></mspace><mo>×</mo><mspace width="3.33333pt"></mspace><msup><mn>10</mn><mn>17</mn></msup></mrow></semantics></math></inline-formula> cm) suggest that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula>-ray flaring event maybe caused by a magnetic reconnection mechanism, but we cannot completely rule out the shock-in-jet model.