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This note considers the computation of the logarithm of symmetric positive definite matrices using the Gauss--Legendre (GL) quadrature. The GL quadrature becomes slow when the condition number of the given matrix is large. In this note, we propose a technique dividing the matrix logarithm into two matrix logarithms, where the condition numbers of the divided logarithm arguments are smaller than that of the original matrix. Although the matrix logarithm needs to be computed twice, each computation can be performed more efficiently, and it potentially reduces the overall computational cost. It is shown that the proposed technique is effective when the condition number of the given matrix is approximately between $130$ and $3.0\times 10^5$.

Interactions between particles normally induce the decay of the particles Bloch oscillations (BOs) in a periodic lattice. In the limit of strong on-site interactions, spin-$1/2$ fermions may form doublon bound states and undergo BOs in the presence of a tilted potential. Here we investigate the impact of nearest-neighbor interaction $V$ on the multi-doublon BOs in a mass-imbalanced extended Fermi-Hubbard model. We derive an effective Hamiltonian for doublons, and show that a slight change in $V$ can qualitatively alter their dynamic behaviors. Notably, at a resonance point, the doublons behave like free hard-core bosons. Under a tilted potential, the system may exhibit different types of multi-doublon BOs at or deviation from the resonance point. Numerical results are presented to demonstrate our conclusions in both one- and two-dimensional systems., Comment: 12 pages, 7 figures

Geometry problem solving (GPS) requires capacities of multi-modal understanding, multi-hop reasoning and theorem knowledge application. In this paper, we propose a neural-symbolic model for plane geometry problem solving (PGPS), named PGPSNet-v2, with three key steps: modal fusion, reasoning process and knowledge verification. In modal fusion, we leverage textual clauses to express fine-grained structural and semantic content of geometry diagram, and fuse diagram with textual problem efficiently through structural-semantic pre-training. For reasoning, we design an explicable solution program to describe the geometric reasoning process, and employ a self-limited decoder to generate solution program autoregressively. To reduce solution errors, a multi-level theorem verifier is proposed to eliminate solutions that do not match geometric principles, alleviating the hallucination of the neural model. We also construct a large-scale geometry problem dataset called PGPS9K, containing fine-grained annotations of textual clauses, solution program and involved knowledge tuples. Extensive experiments on datasets Geometry3K and PGPS9K show that our PGPSNet solver outperforms existing symbolic and neural solvers in GPS performance, while maintaining good explainability and reliability, and the solver components (fusion, reasoning, verification) are all justified effective., Comment: under review by journal

Due to the rapid advancements in multimodal large language models, evaluating their multimodal mathematical capabilities continues to receive wide attention. Despite the datasets like MathVista proposed benchmarks for assessing mathematical capabilities in multimodal scenarios, there is still a lack of corresponding evaluation tools and datasets for fine-grained assessment in the context of K12 education in Chinese language. To systematically evaluate the capability of multimodal large models in solving Chinese multimodal mathematical problems, we propose a Chinese Multi-modal Math Skill Evaluation Benchmark, named CMMaTH, contraining 23k multimodal K12 math related questions, forming the largest Chinese multimodal mathematical problem benchmark to date. CMMaTH questions from elementary to high school levels, provide increased diversity in problem types, solution objectives, visual elements, detailed knowledge points, and standard solution annotations. We have constructed an open-source tool GradeGPT integrated with the CMMaTH dataset, facilitating stable, rapid, and cost-free model evaluation. Our data and code are available.

Two-color laser induced plasma filamentation in air could serve as tabletop sources of broadband terahertz (THz) pulses. Ubiquitous air in the earth facilitates its widespread utilities, particularly, in wireless communication and remote sensing, exploiting the unique advantage of the air-based scheme that the THz sources can be delivered over standoff distances via the pump laser propagation in air. However, the THz emission pattern inevitably has a conical angular profile with a dip in the propagation axis, therefore, THz energy concentration, propagation directionality, and accuracy of signal demodulation are significantly impaired, greatly limiting its direct applications. Here, we successfully eliminate the unfavorable conical profile by experiments and meanwhile enhance THz directionality and intensity by 17 folds by use of abruptly-autofocusing laser beams. Our theory and simulations show that these observations are attributed to efficient suppression of the dephasing effect appearing in previous investigations with ordinary Gaussian laser beams. This scheme is easily accessible since the abruptly-autofocusing beam can be achieved by imposing a spatial light modulator on the input Gaussian beam. This study solves a long-standing problem in the two-color laser scheme and clarifies the underlying physics of the conical angular profile formation., Comment: 22 pages, 4 figures, 1 table

Hierarchical massive quadruple systems are ideal laboratories for examining the theories of star formation, dynamical evolution, and stellar evolution. The successive mergers of hierarchical quadruple systems might explain the mass gap between neutron stars and black holes. Looking for light curves of O-type binaries identified by LAMOST, we find a (2+2) quadruple system: TYC 3340-2437-1, located in the stellar bow-shock nebula (SBN). It has a probability of over 99.99\% being a quadruple system derived from the surface density of the vicinity stars. Its inner orbital periods are 3.390602(89) days and 2.4378(16) days, respectively, and the total mass is about (11.47 + 5.79) + (5.2 + 2.02) = 24.48 $M_{\odot}$. The line-of-sight inclinations of the inner binaries, B$_1$ and B$_2$, are 55.94 and 78.2 degrees, respectively, indicating that they are not co-planar. Based on observations spanning 34 months and the significance of the astrometric excess noise ($D>2$) in Gaia DR3 data, we guess that its outer orbital period might be a few years. If it were true, the quadruple system might form through the disk fragmentation mechanism with outer eccentric greater than zero. This eccentricity could be the cause of both the arc-like feature of the SBN and the noncoplanarity of the inner orbit. The outer orbital period and outer eccentric could be determined with the release of future epoch astrometric data of Gaia.

Within the framework of leading power factorization formalism of nonrelativistic quantum chromodynamics, we calculate the jet fragmentation function for $J/\psi$ production in proton-proton (pp) collisions ranging from $\sqrt{s}=500$ GeV to $13$ TeV. The reasonable agreements between theory and experimental data indicate that $J/\psi$ production within a jet is mainly dominated by gluon fragmentation. Such a mechanism can be further tested by the predicted jet transverse momentum and radius dependence of jet fragmentation function. Based on the satisfying description of pp baseline, we carry out the first theoretical investigation on medium modification on $J/\psi$ production within jet in heavy-ion collisions at the Large Hadron Collider, using a linear Boltzmann transport model combined with hydrodynamics for the simulation of jet-medium interaction. The consistency with the experimental measurement on nuclear modification factor $R_{\text{AA}}$ by CMS collaboration reveals that the gluon jet quenching is the driving force for the suppression of $J/\psi$ production in jet. Furthermore, we make predictions for the dependence of $R_{\text{AA}}$ on the jet transverse momentum and jet radius $R$, which can be tested in future measurements to further constrain the flavor dependence of jet quenching., Comment: 7 pages, 4 figures

In 2016, an exposure meter was installed on the Lijiang Fiber-fed High-Resolution Spectrograph to monitor the coupling of starlight to the science fiber during observations. Based on it, we investigated a method to estimate the exposure flux of the CCD in real time by using the counts of the photomultiplier tubes (PMT) of the exposure meter, and developed a piece of software to optimize the control of the exposure time. First, by using flat-field lamp observations, we determined that there is a linear and proportional relationship between the total counts of the PMT and the exposure flux of the CCD. Second, using historical observations of different spectral types, the corresponding relational conversion factors were determined and obtained separately. Third, the method was validated using actual observation data, which showed that all values of the coefficient of determination were greater than 0.92. Finally, software was developed to display the counts of the PMT and the estimated exposure flux of the CCD in real-time during the observation, providing a visual reference for optimizing the exposure time control., Comment: Published in Research in Astronomy and Astrophysics

Geometry problem solving (GPS) is a challenging mathematical reasoning task requiring multi-modal understanding, fusion, and reasoning. Existing neural solvers take GPS as a vision-language task but are short in the representation of geometry diagrams that carry rich and complex layout information. In this paper, we propose a layout-aware neural solver named LANS, integrated with two new modules: multimodal layout-aware pre-trained language module (MLA-PLM) and layout-aware fusion attention (LA-FA). MLA-PLM adopts structural-semantic pre-training (SSP) to implement global relationship modeling, and point-match pre-training (PMP) to achieve alignment between visual points and textual points. LA-FA employs a layout-aware attention mask to realize point-guided cross-modal fusion for further boosting layout awareness of LANS. Extensive experiments on datasets Geometry3K and PGPS9K validate the effectiveness of the layout-aware modules and superior problem-solving performance of our LANS solver, over existing symbolic and neural solvers. The code will be made public available soon.

We consider the convolution equation $F*X=B$, where $F\in\mathbb{R}^{3\times 3}$ and $B\in\mathbb{R}^{m\times n}$ are given, and $X\in\mathbb{R}^{m\times n}$ is to be determined. The convolution equation can be regarded as a linear system with a coefficient matrix of special structure. This fact has led to many studies including efficient numerical algorithms for solving the convolution equation. In this study, we show that the convolution equation can be represented as a generalized Sylvester equation. Furthermore, for some realistic examples arising from image processing, we show that the generalized Sylvester equation can be reduced to a simpler form, and analyze the unique solvability of the convolution equation.

This paper considers the computation of the matrix exponential $\mathrm{e}^A$ with numerical quadrature. Although several quadrature-based algorithms have been proposed, they focus on (near) Hermitian matrices. In order to deal with non-Hermitian matrices, we use another integral representation including an oscillatory term and consider applying the double exponential (DE) formula specialized to Fourier integrals. The DE formula transforms the given integral into another integral whose interval is infinite, and therefore it is necessary to truncate the infinite interval. In this paper, to utilize the DE formula, we analyze the truncation error and propose two algorithms. The first one approximates $\mathrm{e}^A$ with the fixed mesh size which is a parameter in the DE formula affecting the accuracy. Second one computes $\mathrm{e}^A$ based on the first one with automatic selection of the mesh size depending on the given error tolerance.