論文

A surface model called the fibre bundle model and a 3D object model based on linear Lie algebra model are proposed. Then an algorithm of 3D object recognition using the linear Lie algebra models is presented. It is a convenient recognition method for the objects which are symmetric about some axis. By using the presented algorithm, the representation matrices of the fibre or the base curve from only finite points of the linear Lie algebra model can be obtained. At last some recognition results of practicalities are given. © A paper published in JBIT is protected by copyright.

A novel mathematical model is proposed to provide a comprehensive framework for simulating ephaptic coupling arising from spatiotemporal interactions between propagating neural fiber bundles through the extracellular space. The resulting multidimensional macroscopic model, applicable to arbitrarily shaped fiber bundles, is derived from the Frankenhaeuser-Huxley formulation of neural spike propagation and extends the classic bidomain framework by explicitly accounting for complex fiber bundle geometries, in which intracellular and extracellular domains are nonoverlapping except at the nodes of Ranvier. To enable accurate simulation of such fiber geometries, a high-order continuous Galerkin method is introduced, incorporating anisotropic conductivities and domain discontinuities through orthonormal basis vectors, known as moving frames. The model is validated via two-dimensional simulations of neural spike propagation along both straight and curved fiber bundles.

Molecular dynamics (MD) simulations are powerful tools for elucidating the macroscopic physical properties of materials from microscopic atomic behaviors. However, the massive, high-dimensional data sets generated by MD simulations pose a significant challenge for analysis, necessitating efficient dimensionality reduction and feature extraction techniques. While existing methods such as principal component analysis and unsupervised learning have been utilized, issues regarding data efficiency and computational cost remain. In this study, we propose a statistical analysis framework focusing on the analysis of the particle data distributions through their covariance matrices, corresponding to the second-order moments of MD trajectory data. Discrepancies between system states are quantified using statistical distances between these covariance matrices. By applying dimensionality reduction to the resulting distance matrix, we extract lower-dimensional features that characterize the systems’ dynamics. We validate the proposed method using Lennard-Jones (LJ) particle systems under different temperature conditions, as well as separate bulk systems of ice and liquid water. The results of LJ particles demonstrate an approximately linear correlation between the first principal component obtained through dimensionality reduction of the distance matrix and the diffusion coefficient. This suggests that global physical properties can be effectively inferred from local statistical information, such as covariance matrices, offering a data-efficient alternative for analyzing complex molecular systems. Furthermore, in the case of separate bulk systems of ice and liquid water, the method successfully distinguishes between the two phases, highlighting its potential for characterizing phase transitions and structural differences in molecular systems.

This paper introduces novel approaches and applications for hyper-differential operator-based graph fractional Fourier transform (HGFrFT), which provides a more flexible and efficient framework for graph signal processing. Its mathematical foundation is derived through the relationship between the graph Fourier transform and the fractional graph Fourier transform, thereby enhancing its ability to represent the underlying structure of graph signals. Specifically, four solutions to the Sylvester equation are discussed, and two fixed hyperdifferential operator matrices are introduced to define the HGFrFT. Furthermore, the applications of the HGFrFT in vertex-frequency analysis and optimal filter design are explored. Finally, the effectiveness of the HGFrFT is validated in electrocardiogram signal classification, demonstrating its applicability to real-world signal processing tasks. The theoretical framework and experimental results collectively advance fractional operator theory in graph domains.

Amplifying the separability between the target signal and background clutter constitutes a pivotal approach to advancing the performance of matrix information geometry (MIG) detectors. In order to address target detection within sample-starved heterogeneous clutter environments, this study investigates and applies kernelized MIG detectors. Following standard practice, the sample data is represented as a Hermitian positive-definite (HPD) matrix. The clutter covariance matrix is correspondingly estimated utilizing the geometric mean derived from a collection of secondary HPD matrices. Consequently, the signal detection is framed as distinguishing between two distinct points in the HPD matrix manifold. To augment discriminative capability, HPD matrices undergo transformation into a high-dimensional reproducing kernel Hilbert space via feature maps. This process inherently establishes a positive-definite kernel. We specifically propose three variants of kernelized MIG detectors on HPD manifolds, associated with distinct kernels: the linear kernel, the polynomial kernel, and the radial basis function kernel. Empirical validation on both simulated and real radar datasets confirms the superior performance of the proposed KMIG detectors. Significant advantages are demonstrated relative to both contemporary state-of-the-art alternatives and conventional detectors under conditions of limited samples and nonhomogeneous clutter.

半離散方程式（微分差分方程式ともいう）においては，連続独立変数と離散独立変数が非可換のため，対称性とならない非内在（non-intrinsic）（または不規則（irregular））な変換が生じるという問題があった．本論文では，この問題の解決方法を論じるとともに，半離散方程式における対称性理論について紹介する．具体例として，戸田型，ヴォルテラ型，5 点の伊藤・成田・ボゴヤヴレンスキー方程式などのリー点対称性（Lie point symmetry）を求め，群不変解（group-invariant solution）を導出する．

Diffusion models have emerged as powerful gen-erative models, but their high computational cost in iterative sampling remains a significant bot-tleneck. In this work, we present an in-depth and insightful study of state-of-the-art accelera-tion techniques for diffusion models, including caching and quantization, and reveal their limita-tions in computation error and generation quality. To break these limits, this work introduces Modu-lated Diffusion (MoDiff), an innovative, rigorous, and principled framework that accelerates gener-ative modeling through modulated quantization and error compensation. MoDiff not only inherits the advantages of existing caching and quantiza-tion methods but also serves as a general frame-work to accelerate all diffusion models. The ad-vantages of MoDiff are supported by solid theoret-ical insight and analysis. In addition, extensive ex-periments on CIFAR-10 and LSUN demonstrate that MoDiff significantly reduces activation quan-tization from 8 bits to 3 bits without performance degradation in post-training quantization (PTQ). Our code implementation is available at https://github.com/WeizhiGao/MoDiff.

We construct stochastic multisymplectic systems by considering a stochastic extension to the variational formulation of multisymplectic partial differential equations proposed in Hydon [Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461 (2005): 1627–1637]. The stochastic variational principle implies the existence of stochastic 1-form and 2-form conservation laws, as well as conservation laws arising from continuous variational symmetries via a stochastic Noether's theorem. These results are the stochastic analogs of those found in deterministic variational principles. Furthermore, we develop stochastic structure-preserving collocation methods for this class of stochastic multisymplectic systems. These integrators possess a discrete analog of the stochastic 2-form conservation law and, in the case of linear systems, also guarantee discrete momentum conservation. The effectiveness of the proposed methods is demonstrated through their application to stochastic nonlinear Schrödinger equations featuring either stochastic transport or stochastic dispersion.

This paper proposes a novel graph linear canonical transform framework based on hyper-differential operators, referred to as OGLCT. First, the relationship between the graph fractional Fourier transform (GFrFT) and the graph Fourier transform (GFT) is analyzed, and hyper-differential operators associated with the GFT matrix are derived through solutions to the Sylvester equation. Using these operators, the GFrFT, graph scaling, and graph chirp modulation are defined, culminating in the formal definition of the OGLCT and its fundamental properties. Next, the implementation of the OGLCT is discussed, and parameter optimization techniques are explored. Finally, the application of the OGLCT for feature extraction in image processing is explored, and its effectiveness in image classification tasks is investigated. By adjusting the parameters of the OGLCT, it is demonstrated that flexible tuning significantly enhances model performance, enabling more effective and versatile applications.

Essential characteristics of signal data can be captured by the autocovariance matrix, which, in the stationary scenarios, is Toeplitz Hermitian positive definite (HPD). In this paper, several well-known Riemannian geometric structures of HPD matrix manifolds are applied to signal detection, including the affine invariant Riemannian metric, the log-Euclidean metric, and the Bures–Wasserstein (BW) metric, the last of which was recently extended to HPD manifolds. Riemannian gradient descent algorithms are proposed to solve the corresponding geometric means and medians, that play fundamental roles in the detection process. Simulations within the scenario using the ideal steering vector as the target signal provide compelling evidence that the BW detectors outperform the other geometric detectors as well as the conventional adaptive matched filter and adaptive normalized matched filter when observation data are limited. Further simulations demonstrate that the matrix-CFAR is robust in scenarios where the signal is mismatched. In addition to detection performances, robustness of the geometric detectors to outliers and computational complexity of the algorithms are analysed.