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Large Language Models (LLMs) have made notable progress in mathematical reasoning, yet they often rely on single-paradigm reasoning that limits their effectiveness across diverse tasks. In this paper, we introduce Chain-of-Reasoning (CoR), a novel unified framework that integrates multiple reasoning paradigms--Natural Language Reasoning (NLR), Algorithmic Reasoning (AR), and Symbolic Reasoning (SR)--to enable synergistic collaboration. CoR generates multiple potential answers using different reasoning paradigms and synthesizes them into a coherent final solution. We propose a Progressive Paradigm Training (PPT) strategy that allows models to progressively master these paradigms, culminating in the development of CoR-Math-7B. Experimental results demonstrate that CoR-Math-7B significantly outperforms current SOTA models, achieving up to a 41.0% absolute improvement over GPT-4 in theorem proving tasks and a 7.9% improvement over RL-based methods in arithmetic tasks. These results showcase the enhanced mathematical comprehensive ability of our model, achieving significant performance gains on specific tasks and enabling zero-shot generalization across tasks.

Although fine-tuning Large Language Models (LLMs) with multilingual data can rapidly enhance the multilingual capabilities of LLMs, they still exhibit a performance gap between the dominant language (e.g., English) and non-dominant ones due to the imbalance of training data across languages. To further enhance the performance of non-dominant languages, we propose ShifCon, a Shift-based Contrastive framework that aligns the internal forward process of other languages toward that of the dominant one. Specifically, it shifts the representations of non-dominant languages into the dominant language subspace, allowing them to access relatively rich information encoded in the model parameters. The enriched representations are then shifted back into their original language subspace before generation. Moreover, we introduce a subspace distance metric to pinpoint the optimal layer area for shifting representations and employ multilingual contrastive learning to further enhance the alignment of representations within this area. Experiments demonstrate that our ShifCon framework significantly enhances the performance of non-dominant languages, particularly for low-resource ones. Further analysis offers extra insights to verify the effectiveness of ShifCon and propel future research, Comment: 23 pages, 11 figures

Primordial black holes (PBHs) are considered viable candidates for dark matter and the seeds of supermassive black holes (SMBHs), with their fruitful physical influences providing significant insights into the conditions of the early Universe. Cosmic microwave background (CMB) $\mu$ distortion tightly constrain the abundance of PBHs in the mass range of $10^4 \sim 10^{11} M_{\odot}$ recently, limiting their potential to serve as seeds for the SMBHs observed. Given that $\mu$ distortion directly constrain the primordial power spectrum, it is crucial to employ more precise methods in computing PBH abundance to strengthen the reliability of these constraints. By a Press-Schechter (PS) type method utilizing the compaction function, we find that the abundance of PBHs could be higher than previously estimated constraints from $\mu$ distortion observations. Furthermore, our analysis shows that variations in the shape of the power spectrum have a negligible impact on our conclusions within the mass ranges under consideration. This conclusion provides us a perspective for further research on the constrain of PBH by $\mu$ distortion., Comment: 9 pages, 3 figures

Extriangulated categories, introduced by Nakaoka and Palu, serve as a simultaneous generalization of exact and triangulated categories. In this paper, we first introduce the concept of admissible weak factorization systems and establish a bijection between cotorsion pairs and admissible weak factorization systems in extriangulated categories. Consequently, we give the equivalences between hereditary cotorsion pairs and compatible cotorsion pairs via admissible weak factorization systems under certain conditions in extriangulated categories, thereby generalizing a result by Di, Li, and Liang., Comment: 13 pages

Cosmic birefringence, the observed rotation of the polarization plane of the cosmic microwave background (CMB), serves as a compelling probe for parity-violating physics beyond the Standard Model. This study explores the potential of ultralight axion-like particle (ALP) dark matter to explain the observed cosmic birefringence in the CMB. We focus on the previously understudied mass range of $10^{-25}$ eV to $10^{-23}$ eV, where ALPs start to undergo nonlinear clustering in the late universe. Our analysis incorporates recent cosmological constraints and considers the washout effect on CMB polarization. We find that for models with ALP masses $10^{-25}$ eV $\lesssim m_\phi \lesssim 10^{-23}$ eV and birefringence arising from late ALP clustering, the upper limit on the ALP-photon coupling constant, imposed by the washout effect, is stringently lower than the coupling required to account for the observed static cosmic birefringence signal. This discrepancy persists regardless of the ALP fraction in dark matter. Furthermore, considering ALPs with masses $m_\phi\gtrsim$ $10^{-23}$ eV cannot explain static birefringence due to their rapid field oscillations, our results indicate that, all ALP dark matter candidates capable of nonlinear clustering in the late universe and thus contributing mainly to the rotation angle of polarized photons, are incompatible with explaining the static cosmic birefringence signal observed in Planck and WMAP data.

Present cosmic microwave background (CMB) observations have significantly advanced our understanding of the universe's origin, especially with primordial gravitational waves (PGWs). Currently, ground-based CMB telescopes are mainly located in the southern hemisphere, leaving an untapped potential for observations in the northern hemisphere. In this work, we investigate the perspective of a northern hemisphere CMB polarization telescope (NHT) to detect PGWs and present mock data for such a project. We forecast the detection sensitivity on the tensor-to-scalar ratio r of NHT and compare it with the existed ground-based experiments, also search for optimal experimental configurations that can achieve the best sensitivity of r. Our results indicate that, considering realistic experimental conditions, the first year of NHT observations combined with Planck can achieve a precision of \sigma (r)= 0.015, reaching the level of BICEP2/Keck, with significant potential for improvement with subsequent instrumentation parameter enhancements.

Fuzzy dark matter (FDM) with mass around $10^{-22}$ eV is viewed as a promising paradigm in understanding the structure formation of the local universe at small scales. Recent observations, however, begin to challenge FDM in return. We focus on the arguments between the solution to CDM small-scale curiosities and recent observations on matter power spectrum, and find its implication on an earlier formation of small-scale structure. In this article, we propose a scheme of k-ULDM scalar field with a differently-evolving sound speed, thanks to the non-canonical kinetics. With the help of the Dirac-Born-Infeld (DBI) theory, we illustrate to change the behavior of the quantum pressure term countering collapse, therefore change the history of structure growth. We find that it can truly reopen the ULDM mass window closed by the Lyman-$\alpha$ problem. We will discuss such examples in this paper, while more possibilities remain to be explored., Comment: 25 pages, 10 figures

Context. The extreme precision and accuracy of forthcoming observations of CMB temperature and polarization anisotropies, aiming to detect the tiny signatures of primordial gravitational waves or of light relic particles beyond the standard three light neutrinos, requires commensurate precision in the modelling of foreground Galactic emission that contaminates CMB observations. Aims. We evaluate the impact of second-order effects in Galactic foreground emission due to Thomson scattering off interstellar free electrons and to Rayleigh scattering off interstellar dust particles. Methods. We use existing sky survey data and models of the distribution of free electrons and dust within the Milky Way to estimate the amplitude and power spectra of the emission originating from radiation scattered either by free electrons or by dust grains at CMB frequencies. Results. Both processes generate corrections to the total emission that are small compared to direct emission, and are small enough not to pose problems for current-generation observations. Conclusions. However, B-modes generated by Thomson scattering of incoming radiation by interstellar free electrons at CMB frequencies are within an order of magnitude of the sensitivity of the most advanced forthcoming CMB telescopes, and might require more precise evaluation in the future., Comment: 8 pages, 5 figures

Let $\mathcal{C}$ be a weakly idempotent complete extriangulated category. In contrast with the Hovey correspondence of admissible model structures on weakly idempotent complete exact categories from two complete cotorsion pairs, we give a construction of model structures on $\mathcal{C}$ from only one complete cotorsion pair. Our main result not only generalizes the work by Beligiannis-Reiten and Cui-Lu-Zhang, but also provides methods to construct model structures from silting objects of $\mathcal{C}$ and co-$t$-structures in triangulated categories., Comment: 23 pages

We perform a cosmological test of Cotton gravity, which describes gravity by cotton tensor. We assume in Cotton gravity the background evolution is the same as the flat FLRW background. We derive the cosmological perturbation theory of the scalar mode at the linear level, where the difference from the $\Lambda$CDM model is characterized by the parameter $\beta$. We incorporate Cotton gravity with a neutrino model and perform a Monte Carlo Markov Chain (MCMC) analysis using data from the Cosmic Microwave Background (CMB) and Sloan Digital Sky Survey (SDSS). The analysis constrains parameter $\beta=-0.00008^{+0.00080}_{-0.00104}$ at the 1-$\sigma$ confidence level. We conclude that currently, there is no obvious deviation between Cotton gravity and the $\Lambda$CDM model in the linear cosmological perturbation level for observations., Comment: 13 pages, 15 figures

We reconstruct the cosmological background evolution under the scenario of dynamical dark energy through the Gaussian process approach, using the latest Dark Energy Spectroscopic Instrument (DESI) baryon acoustic oscillations (BAO) combined with other observations. Our results reveal that the reconstructed dark-energy equation-of-state (EoS) parameter $w(z)$ exhibits the so-called quintom-B behavior, crossing $-1$ from phantom to quintessence regime as the universe expands. We investigate under what situation this type of evolution could be achieved from the perspectives of field theories and modified gravity. In particular, we reconstruct the corresponding actions for $f(R)$, $f(T)$, and $f(Q)$ gravity, respectively. We explicitly show that, certain modified gravity can exhibit the quintom dynamics and fit the recent DESI data efficiently, and for all cases the quadratic deviation from the $\Lambda$CDM scenario is mildly favored., Comment: 12 pages, 3 figures

Large language models (LLMs) demonstrate remarkable multilingual capabilities without being pre-trained on specially curated multilingual parallel corpora. It remains a challenging problem to explain the underlying mechanisms by which LLMs process multilingual texts. In this paper, we delve into the composition of Transformer architectures in LLMs to pinpoint language-specific regions. Specially, we propose a novel detection method, language activation probability entropy (LAPE), to identify language-specific neurons within LLMs. Based on LAPE, we conduct comprehensive experiments on several representative LLMs, such as LLaMA-2, BLOOM, and Mistral. Our findings indicate that LLMs' proficiency in processing a particular language is predominantly due to a small subset of neurons, primarily situated in the models' top and bottom layers. Furthermore, we showcase the feasibility to "steer" the output language of LLMs by selectively activating or deactivating language-specific neurons. Our research provides important evidence to the understanding and exploration of the multilingual capabilities of LLMs., Comment: Accepted by ACL 2024

We introduce Generalized Instruction Tuning (called GLAN), a general and scalable method for instruction tuning of Large Language Models (LLMs). Unlike prior work that relies on seed examples or existing datasets to construct instruction tuning data, GLAN exclusively utilizes a pre-curated taxonomy of human knowledge and capabilities as input and generates large-scale synthetic instruction data across all disciplines. Specifically, inspired by the systematic structure in human education system, we build the taxonomy by decomposing human knowledge and capabilities to various fields, sub-fields and ultimately, distinct disciplines semi-automatically, facilitated by LLMs. Subsequently, we generate a comprehensive list of subjects for every discipline and proceed to design a syllabus tailored to each subject, again utilizing LLMs. With the fine-grained key concepts detailed in every class session of the syllabus, we are able to generate diverse instructions with a broad coverage across the entire spectrum of human knowledge and skills. Extensive experiments on large language models (e.g., Mistral) demonstrate that GLAN excels in multiple dimensions from mathematical reasoning, coding, academic exams, logical reasoning to general instruction following without using task-specific training data of these tasks. In addition, GLAN allows for easy customization and new fields or skills can be added by simply incorporating a new node into our taxonomy., Comment: Work in progress

In light of the non-perturbative resonance effects that may occur during inflation, we introduce a parametrization for the power spectrum of the stochastic gravitational wave background (SGWB) characterized by narrow-band amplification. We utilize the universal $\Omega_\text{GW}\propto k^3$ infrared limit, applicable to a wide array of gravitational wave sources, to devise a robust yet straightforward parameterization optimized for Markov Chain Monte Carlo (MCMC) analyses. This parameterization is demonstrated through select examples where its application is pertinent, and we discuss the advantages of this approach over traditional parametrizations for narrow-band scenarios. To evaluate the sensitivity of our proposed model parameters, we apply a mock likelihood based on the CMB-Stage4 data. Furthermore, we explicate the computational process for the mapping relationship between the foundational model parameters and our parameterized framework, using a two-field inflation model that resonantly amplifies gravitational waves (GWs) as an example.

We are studying the effects of Self-Interacting dark radiation (SIdr) on the evolution of the universe. Our main focus is on the cosmic microwave background (CMB) and how SIdr could potentially help resolve the Hubble tension. We are looking into different scenarios by mixing SIdr with Free-Streaming dark radiation (FSdr) or not to determine whether SIdr can indeed contribute to solving the Hubble tension. We find that SIdr alone can increase the Hubble constant ($H_0$) to $70.1_{-1.6}^{+1.3}, \text{km/s/Mpc}$ with a value of $N_{\rm eff}=3.27_{-0.31}^{+0.23}$. However, including \lzy{FSdr} disfavors the existence of SIdr $\tilde N_{\rm si}\approx0.37$. Even though the Hubble constant is increased compared to the predicted value, it entails $N_{\rm eff}=3.52\pm0.25$. Finally, we implement the Fisher method for future experiments and a $7.64\sigma$ measurement of $\tilde N_{\rm si}$ will be obtained when combing data from Planck, AliCPT, and CMB-S4., Comment: 29 pages, 13 figures, 5 tables