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Advancements in LLMs have significantly expanded their capabilities across various domains. However, mathematical reasoning remains a challenging area, prompting the development of math-specific LLMs. These models typically follow a two-stage training paradigm: pre-training with math-related corpora and post-training with problem datasets for SFT. Despite these efforts, the improvements in mathematical reasoning achieved through continued pre-training (CPT) are often less significant compared to those obtained via SFT. This study addresses this discrepancy by exploring alternative strategies during the pre-training phase, focusing on the use of problem-solving data over general mathematical corpora. We investigate three primary research questions: (1) Can problem-solving data enhance the model's mathematical reasoning capabilities more effectively than general mathematical corpora during CPT? (2) Are synthetic data from the same source equally effective, and which synthesis methods are most efficient? (3) How do the capabilities developed from the same problem-solving data differ between the CPT and SFT stages, and what factors contribute to these differences? Our findings indicate that problem-solving data significantly enhances the model's mathematical capabilities compared to general mathematical corpora. We also identify effective data synthesis methods, demonstrating that the tutorship amplification synthesis method achieves the best performance. Furthermore, while SFT facilitates instruction-following abilities, it underperforms compared to CPT with the same data, which can be partially attributed to its poor learning capacity for hard multi-step problem-solving data. These insights provide valuable guidance for optimizing the mathematical reasoning capabilities of LLMs, culminating in our development of a powerful mathematical base model called JiuZhang-8B., Comment: ICLR 2025

The quantum spin Hall effect has been predicted in twisted homobilayer transition metal dichalcogenides (TMDs) owing to the layer-pseudospin magnetic field. Recently, experimental observations have also confirmed such topological states of matter. However, the topological electronic properties in multilayer moir\'e superlattices remain to be further explored. In twisted TMDs heterotrilayers, the realization of moir\'e potential with various symmetries becomes feasible. Here, we demonstrate that twisted trilayer TMDs can enter a topological insulator phase under the influence of moir\'e potential with ${C_6}$ symmetry. Specifically, we built two types of trilayer heterostructures, where the low-energy valence band electrons are contributed by the middle layer. In the AA-stacked moir\'e WS$_2$/WSe$_2$/MoS$_2$ heterotrilayers where only the middle layer is twisted, the maxima of the moir\'e potential exhibits an approximate ${C_6}$ symmetry. The $C_6$ symmetry effectively compensates for the spatial inversion symmetry breaking in the WSe$_2$ layer, leading to a twist-angle-dependent topological phase transition. Leveraging a Green's function approach, we calculate the local state density of edge states at topological minigaps, confirming their nature as moir\'e edge states. In the helical twisted AA-stacked moir\'e MoS$_2$/WSe$_2$/MoS$_2$ heterotrilayers, we observed a mosaic pattern of topological and trivial insulators. The emergence of topological mosaic is attributed to the maxima of the local moir\'e potential possessing $C_6$ symmetry. The results provide a new way for the experimental realization of topological phases in TMDs heterojunctions., Comment: 8 pages, 6 figures

Typically, the significant efficiency can be achieved by deploying different edge AI models in various real world scenarios while a few large models manage those edge AI models remotely from cloud servers. However, customizing edge AI models for each user's specific application or extending current models to new application scenarios remains a challenge. Inappropriate local training or fine tuning of edge AI models by users can lead to model malfunction, potentially resulting in legal issues for the manufacturer. To address aforementioned issues, this paper proposes an innovative framework called "DiReD", which involves knowledge DIstillation & REverse DIstillation. In the initial step, an edge AI model is trained with presumed data and a KD process using the cloud AI model in the upper management cloud server. This edge AI model is then dispatched to edge AI devices solely for inference in the user's application scenario. When the user needs to update the edge AI model to better fit the actual scenario, the reverse distillation (RD) process is employed to extract the knowledge: the difference between user preferences and the manufacturer's presumptions from the edge AI model using the user's exclusive data. Only the extracted knowledge is reported back to the upper management cloud server to update the cloud AI model, thus protecting user privacy by not using any exclusive data. The updated cloud AI can then update the edge AI model with the extended knowledge. Simulation results demonstrate that the proposed "DiReDi" framework allows the manufacturer to update the user model by learning new knowledge from the user's actual scenario with private data. The initial redundant knowledge is reduced since the retraining emphasizes user private data.

Electric dark spots are point singularities at which the electric field amplitude vanishes. These singularities usually emerge in real space accidentally and are unstable due to the vectorial property of the electric field. In this paper, we show that topologically protected electric dark spots can emerge in metal scatterers under external excitation. The material property of metal imposes a boundary condition that reduces the vectorial electric field on the metal surface to a scalar field. The phase singularity of this scalar field has zero amplitude and carries a well-defined topological charge corresponding to an electric dark spot. The topological electric dark spots give rise to the superoscillation phenomenon with a divergent local wavenumber. We uncover the global charge conservation property of the dark spots on the scatterers' surfaces and demonstrate their stability under different perturbations. We also demonstrate the manipulation of the dark spots' topological charge and spatial location. The results open a new avenue for nanophotonic near-field manipulations and may find applications in optical metrology, optical sensing, and super-resolution imaging., Comment: 12 pages, 6 figures

Artificial structures have been widely used to manipulate sound fields. Most properties of these structures derive from the material and geometry. Few are explicitly related to the structural topology in the real space. Here, we discover a fundamental connection between the real-space topology of acoustic structures and the topological properties of sound fields. We find that the genus of acoustic cavities can protect the birth of singularities in velocity polarization and isopressure line fields. The total topological index of the surface singularities is equal to the cavities' Euler characteristic. This intriguing relationship is rooted in the Poincare-Hopf theorem and is irrelevant to the specific material, geometric details, or excitation properties. The isopressure line singularities lead to acoustic hotspots and quiet zones. The velocity polarization singularities give rise to nontrivial polarization Mobius strips and skyrmion textures. The results enable sound sculpturing with structural topology and can find applications in acoustic communications, acoustic sensing, and noise control., Comment: 26 pages, 11 figures

Circular dichroism (CD) is an intriguing chiroptical phenomenon associated with the interaction of chiral structures with circularly polarized lights. Although the CD effect has been extensively studied in optics, it has not yet been demonstrated in acoustic systems. Here, we demonstrate the acoustic CD effect in a three-dimensional chiral metamaterial supporting circularly polarized transverse sound. We find that the effect is negligible in the lossy metamaterial possessing $C_4$ rotational symmetry but can be strongly enhanced in the $C_2$-symmetric system with inhomogeneous loss. The phenomena can be understood based on the properties of the metamaterial's complex band structure and the quality factors of its eigenmodes. We show that the enhanced CD in the $C_2$-symmetric system is attributed to the polarization bandgaps and the non-Hermitian exceptional points appearing near the Brillouin-zone center and boundaries. The results contribute to the understanding of chiral sound-matter interactions and can find applications in acoustic sensing of chiral structures and sound manipulations based on its vector properties., Comment: 9 pages, 9 figures

In long-wavelength moir\'e superlattices of stacked transition metal dichalcogenides (TMDs), structural reconstruction ubiquitously occurs, which has reported to impact significantly their electronic properties. However, complete microscopic understandings of the interplay between the lattice reconstruction and alteration of electronic properties, and their further response to external perturbations in the reconstructed TMDs moir\'e superlattice are still lacking. Here, using scanning tunneling microscopy (STM) and scanning tunneling spectroscopy (STS) combined with first-principles calculation, we study the strain-dependent structural reconstruction and its correlated electronic reconstruction in long-wavelength H-type WS$_{2}$ moir\'e superlattice at nanometer scale. We observe that the long-wavelength WS$_{2}$ moir\'e superlattices experiencing strong atomic reconstruction transform into a hexagonal array of screw dislocations separating large-sized H-stacked domains. Both the geometry and the moir\'e wavelength of the moir\'e superlattice are dramatically tuned by external intralayer heterostrain in our experiment. Remarkably, the STS measurements further demonstrate that the location of the K point in conduction band is modulated sensitively by strain-induced lattice deformation at nanometer scale in this system, with the maximum energy shift reaching up to 300 meV. Our results highlight that intralayer strain plays a vital role in determining structural and electronic properties in TMD moir\'e superlattice.

Circular dichroism and helical dichroism are intriguing chiroptical phenomena with broad applications in optical sensing and imaging. Here, we generalize one of the phenomena-helical dichroism-to acoustics. We show that a one-dimensional lattice of chiral resonators with loss can induce differential absorption of helical sounds (i.e. acoustic vortices) carrying opposite orbital angular momentum (OAM). This acoustic helical dichroism strongly depends on the rotation symmetry of the chiral resonators. A breaking of the $C_4$ rotation symmetry can induce coupling between the opposite chiral dipole modes of the resonators. This leads to OAM bandgaps and non-Hermitian exceptional points near the Brillouin-zone center and boundaries, which together give rise to significantly enhanced helical dichroism. The underlying physics can be well captured by an effective Hamiltonian that quantitatively reproduces the complex band structures. The acoustic helical dichroism can find important applications in acoustic OAM manipulations and chiral sound-matter interactions., Comment: 9 pages, 7 figures

Spin-orbit interactions (SOIs) endow light with intriguing properties and applications such as photonic spin-Hall effects and spin-dependent vortex generations. However, it is counterintuitive that SOIs can exist for sound, which is a longitudinal wave that carries no intrinsic spin. Here, we theoretically and experimentally demonstrate that airborne sound can possess artificial transversality in an acoustic micropolar metamaterial and thus carry both spin and orbital angular momentum. This enables the realization of acoustic SOIs with rich phenomena beyond those in conventional acoustic systems. We demonstrate that acoustic activity of the metamaterial can induce coupling between the spin and linear crystal momentum k, which leads to negative refraction of the transverse sound. In addition, we show that the scattering of the transverse sound by a dipole particle can generate spin-dependent acoustic vortices via the geometric phase effect. The acoustic SOIs can provide new perspectives and functionalities for sound manipulations beyond the conventional scalar degree of freedom and may open an avenue to the development of spin-orbit acoustics., Comment: 31 pages, 8 figures