Two-dimensional representations of the genus two surface group

Let $\Pi$ denote the fundamental group of the closed surface of genus 2. For any quadratically closed ring $R$ with $2$ invertible, we classify irreducible representations $\Pi\to{\rm SL}(2,R)$ up to conjugacy by giving explicit formulas.Comment: 6 page

Breaking Language Barriers in Multilingual Mathematical Reasoning: Insights and Observations

Existing research predominantly focuses on developing powerful language learning models (LLMs) for mathematical reasoning within monolingual languages, with few explorations in preserving efficacy in a multilingual context. To bridge this gap, this paper pioneers exploring and training powerful Multilingual Math Reasoning (xMR) LLMs. Firstly, by utilizing translation, we construct the first multilingual math reasoning instruction dataset, MGSM8KInstruct, encompassing ten distinct languages, thus addressing the issue of training data scarcity in xMR tasks. Based on the collected dataset, we propose different training strategies to build powerful xMR LLMs, named MathOctopus, notably outperform conventional open-source LLMs and exhibit superiority over ChatGPT in few-shot scenarios. Notably, MathOctopus-13B reaches 47.6% accuracy which exceeds ChatGPT 46.3% on MGSM testset. Beyond remarkable results, we unearth several pivotal observations and insights from extensive experiments: (1) When extending the rejection sampling strategy to the multilingual context, it proves effective for model performances, albeit limited. (2) Employing parallel corpora for math Supervised Fine-Tuning (SFT) across multiple languages not only significantly enhances model performance multilingually but also elevates their monolingual performance. This indicates that crafting multilingual corpora can be regarded as a vital strategy for enhancing model performance in a specific language, especially in mathematical reasoning tasks. For instance, MathOctopus-7B improves its counterparts that trained on English from 42.2% to 50.8% on GSM8K testset.Comment: Work in Progres

Nickel hydroxide-supported ru single atoms and Pd nanoclusters for enhanced electrocatalytic hydrogen evolution and ethanol oxidation

The rational fabrication of Pt-free catalysts for driving the development of practical applications in alkaline water electrolysis and fuel cells is promising but challenging. Herein, a promising approach is outlined for the rational design of multimetallic catalysts comprising multiple active sites including Pd nanoclusters and Ru single atoms anchored at the defective sites of Ni(OH)2 to simultaneously enhance hydrogen evolution reactions (HER) and ethanol oxidation reactions (EOR). Remarkably, Pd12Ru3/Ni(OH)2/C exhibits a remarkably reduced HER overpotential (16.1 mV@10 mA cm−2 with a Tafel slope of 21.8 mV dec−1) as compared to commercial 20 wt.% Pt/C (26.0 mV@10 mA cm−2, 32.5 mV dec−1). More importantly, Pd12Ru3/Ni(OH)2/C possesses a self-optimized overpotential to 12.5 mV@10 mA cm−2 after 20 000 cycles stability test while a significantly decreased performance for commercial 20wt.% Pt/C (64.5 mV@10 mA cm−2 after 5000 cycles). The mass activity of Pd12Ru3/Ni(OH)2/C for the EOR is up to 3.724 A mgPdRu−1, ≈20 times higher than that of commercial Pd/C. Electrochemical in situ Fourier transform infrared measurements confirm the enhanced CO2 selectivity of Pd12Ru3/Ni(OH)2/C while synergistic and electronic effects of adjacent Ru, Pd, and OHad adsorption on Ni(OH)2 at low potential play a key role during EOR

64-fs L-band pulse generation by an all-fibre Er-doped laser

We demonstrate a L-band all-fibre erbium-doped laser mode-locked by nonlinear polarisation rotation. The use of a single gain segment with appropriate length and dispersion and a L-band optimised Brewster fibre grating as an in-fibre polariser enables the generation of 64-fs pulses at 1.59µm

Generation of 64-fs L-band stretched pulses from an all-fibre Er-doped laser

We demonstrate an L-band all-fibre erbium-doped laser mode locked by nonlinear polarisation rotation and working in the stretched-pulse regime. The use of a single segment of gain fibre with appropriate length and dispersion and a Brewster fibre grating optimised for the L band as an in-fibre polariser enables the generation of pulses at 1.59-μm central wavelength, which can be linearly compressed to 64-fs duration. Numerical simulations of the laser model support our experimental findings. Our laser design gives a route towards low-cost and low-complexity fibre-integrated laser sources for applications requiring L-band ultrashort pulses

Number of Unfavorable Intermediate‐Risk Factors Predicts Pathologic Upstaging and Prostate Cancer‐Specific Mortality Following Radical Prostatectomy: Results From the SEARCH Database

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135480/1/pros23255.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/135480/2/pros23255_am.pd