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The Split Quaternion Least Squares Problem With Equality Constraint and Its Application

MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2025年

作者： Zhou, Zhi-Han Meng, Zi-Xiang Li, Ying Zhang, Feng-Xia Liaocheng Univ Coll Math & Sci Liaocheng Peoples R China

By applying the real representation matrix and the Moore-Penrose inverse of the split quaternion matrix, we first study the split quaternion least squares problem with equality constraint (SQLSE) and obtain its general solution and minimum norm solution. Then, we obtain the Karush-Kuhn-Tucker equation, which is an equivalent problem of the SQLSE problem. Immediately after, we propose a numerical algorithm on the real number field for the minimum norm solution of the SQLSE problem and enumerate three examples to display its validity. Eventually, we provide an application of the SQLSE problem in the color image restoration.

关键词： color image restoration equality constraint numerical algorithm real representation matrix split quaternion least squares problem

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Does the nonuniqueness of the discrete relaxation spectrum really matter?

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RHEOLOGICA ACTA 2025年第2-3期64卷 107-116页

作者： Shanbhag, Sachin Florida State Univ Dept Sci Comp Tallahassee FL 32306 USA

Unlike the continuous relaxation spectrum (CRS), discrete relaxation spectra (DRS) are nonunique. This means that the linear viscoelastic response of a material can be described by two or more distinct DRS. Constraints like parsimony and consistency help us to infer meaningful DRS, but are not sufficient to induce uniqueness as it is an inherent property of discretization. Using parsimonious DRS from two different programs (DISCRETE and pyReSpect) on data drawn from experiments, simulations, and theory, we demonstrate that nonuniqueness does not hinder the two most common applications of relaxation spectra, viz. characterization and interconversion. Furthermore, information for reconstructing the CRS underlying the data is embedded in the DRS. Therefore, for most practical considerations, we find that the nonuniqueness of the DRS does not matter.

关键词： Relaxation spectrum numerical algorithm Linear viscoelasticity

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Efficient Lie derivative algorithm for two special nonlinear equations

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PRAMANA-JOURNAL OF PHYSICS 2024年第3期98卷 1-10页

作者： Gu, Zhi-Hang Jiang, Wen-An Chen, Li-Qun Jiangsu Univ Fac Civil Engn & Mech Zhenjiang 212013 Peoples R China Shanghai Univ Dept Mech Shanghai 200000 Peoples R China

This paper explores the effectiveness of the Lie derivative discretisation scheme applied to two particular types of nonlinear dynamical equations, both of which have the characteristic of time variables in the denominator position. The discrete structure of non-autonomous systems is established. In particular, we exclude time variables as state variables to prevent non-autonomous systems from becoming autonomous systems. Using this method, we compute the numerical solution of the system above and compare it with the precise solution and the numerical findings of Runge-Kutta, demonstrating the broad applicability of the Lie derivative numerical algorithm. Finally, we determine the CPU consumption time of two numerical algorithms, thus providing evidence of the high efficiency of the Lie derivative numerical algorithm.

关键词： Lie derivative numerical algorithm high efficiency nonlinear dynamical equations 12.60.Jv 12.10.Dm 98.80.Cq 11.30.Hv

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Modeling and experimental validation of the contact characteristics for laying during automated fiber placement

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INTERNATIONAL JOURNAL OF ADVANCED MANUFACTURING TECHNOLOGY 2024年第12期131卷 5997-6011页

作者： Li, Yan He, Yuxiao Jiang, Junxia Wang, Han Zhu, Weidong Wang, Qing Cheng, Liang Zhejiang Univ Sch Mech Engn State Key Lab Fluid Power & Mechatron Syst Hangzhou 310027 Peoples R China Zhejiang Univ Sch Mech Engn Key Lab Adv Mfg Technol Zhejiang Prov Hangzhou 310027 Peoples R China Aerosp Syst Engn Shanghai Shanghai 201100 Peoples R China

In this paper, a theoretical model of the contact characteristics for layers on irregular surfaces is established by analyzing the contact deformation and slip ratio at the placement path points. Based on this model, a numerical algorithm for calculating the contact characteristics at any path point is proposed. The normal stress field, tangential stress field, and contact status of the contact area at four path points on the double-curved surface mold are obtained using the algorithm and verified through experiments. The results indicate that the model can be used to predict the contact characteristics at any placement path point, and the possible defects at points P3 and P4 are initially identified by analyzing the contact characteristics. In addition, compared to the 8-h computational time required for simulation calculation, this numerical algorithm takes only 20 ms to compute, which shows that the developed model has greatly reduced the calculation time.

关键词： Automated fiber placement Contact characteristics numerical algorithm Laying process

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Representation and regression problems in neural networks: Relaxation, generalization, and numerics

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MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES 2025年第6期35卷 1471-1521页

作者： Liu, Kang Zuazua, Enrique Univ Bourgogne Europe Inst Math Bourgogne CNRS F-21000 Dijon France Friedrich Alexander Univ Erlangen Nurnberg Dept Math D-91058 Erlangen Germany Univ Autonoma Madrid Dept Matemat Madrid 28049 Spain Fdn Deusto Bilbao Bilbao 48007 Basque Country Spain

In this work, we address three non-convex optimization problems associated with the training of shallow neural networks (NNs) for exact and approximate representation, as well as for regression tasks. Through a mean-field approach, we convexify these problems and, applying a representer theorem, prove the absence of relaxation gaps. We establish generalization bounds for the resulting NN solutions, assessing their predictive performance on test datasets, analyzing the impact of key hyperparameters on these bounds, and proposing optimal choices. On the computational side, we examine the discretization of the convexified problems and derive convergence rates. For low-dimensional datasets, these discretized problems are efficiently solvable using the simplex method. For high-dimensional datasets, we propose a sparsification algorithm that, combined with gradient descent for over-parameterized shallow NNs, yields effective solutions to the primal problems.

关键词： Neural network mean-field relaxation representer theorem generalization numerical algorithm

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Efficient simulation of mixed boundary value problems and conformal mappings

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APPLIED MATHEMATICS AND COMPUTATION 2025年 488卷

作者： Han, Qiansheng Rasila, Antti Sottinen, Tommi Guangdong Technion Israel Inst Technol Dept Math Comp Sci Shantou 515063 Guangdong Peoples R China Technion Israel Inst Technol Dept Math IL-32000 Haifa Israel Univ Vaasa Sch Technol & Innovat POB 700 FIN-65101 Vaasa Finland

We present a stochastic method for the simulation of Laplace's equation with a mixed boundary condition in planar domains that are polygonal or bounded by circular arcs. We call this method the Reflected Walk-on-Spheres algorithm. The method combines a traditional Walk-on- Spheres algorithm with use of reflections at the Neumann boundaries. We apply our algorithm to simulate numerical conformal mappings from certain quadrilaterals to the corresponding canonical domains, and to compute their conformal moduli. Finally, we give examples of the method on three dimensional polyhedral domains, and use it to simulate the heat flow on an Lshaped insulated polyhedron.

关键词： Brownian motion Monte Carlo methods Walk on spheres Harmonic measure Conformal mappings Conformal modulus numerical algorithm Heat flow

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Description of the Lubricant Behavior Based on the Theory of Elasto-Viscoplastic

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MATERIALS 2025年第6期18卷 1360-1360页

作者： Nosov, Yuriy O. Kamenskikh, Anna A. Bogdanova, Anastasia P. Perm Natl Res Polytech Univ Dept Computat Math Mech & Biomech Perm 614990 Russia

The aim of the work is to provide a mathematical description of the lubricant's behavior model used in sliding bearings of bridge structures. It was previously established that the Maxwell model does not correctly describe the lubricant's behavior in a wide range of temperatures and deformation rates. The lubricant model should take into account not only viscosity but also plasticity. The Anand model, which was adapted by introducing temperature dependencies for a number of material parameters, was chosen to describe the lubricant behavior. The functionality of the previously created procedure for identifying material properties was also expanded on the modified Anand model. This made it possible to obtain a lubricant mathematical model with an error of less than 5% in the operating temperature range from -40 to +80 degrees C. The study included a description of the behavior model for two lubricants: CIATIM-221 and CIATIM-221F. CIATIM-221F differs from CIATIM-221 by including superfine particles of polytetrafluoroethylene (PTFE) to improve properties. The study confirmed that the modified Anand model allows describing the material behavior more accurately than the Maxwell model. It was found that the samples behave as a solid over the entire temperature range (from -40 to +80 degrees C). A comparative analysis of the thermal behavior of CIATIM-221 and CIATIM-221F was performed.

关键词： numerical algorithm mathematical model Anand model lubricant thermophysical properties computer-aided design systems

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Modelling the transmission behavior of measles disease considering contaminated environment through a fractal-fractional Mittag-Leffler kernel

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PHYSICA SCRIPTA 2024年第7期99卷 075025-075025页

作者： Wireko, Fredrick A. Adu, Isaac K. Gyamfi, Kwame A. Asamoah, Joshua Kiddy K. Kwame Nkrumah Univ Sci & Technol Dept Math Kumasi Ghana Kumasi Tech Univ Dept Math Sci Kumasi Ghana Saveetha Sch Engn SIMATS Dept Math Chennai India

This work utilises a fractal-fractional operator to examine the dynamics of transmission of measles disease. The existence and uniqueness of the measles model have been thoroughly examined in the context of the fixed point theorem, specifically utilising the Atangana-Baleanu fractal and fractional operators. The model has been demonstrated to possess both Hyers-Ulam stability and Hyers-Ulam Rassias stability. Furthermore, a qualitative analysis of the model was performed, including examination of key parameters such as the fundamental reproduction number, the measles-free and measles-present equilibria, and assessment of global stability. This research has shown that the transmission of measles disease is affected by natural phenomena, as changes in the fractal-fractional order lead to changes in the disease dynamics. Furthermore, environmental contamination has been shown to play a significant role in the transmission of the measles disease.

关键词： measles Newton's approximation technique fractal-fractional derivative numerical algorithm

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When can forward stable algorithms be composed stably?

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IMA JOURNAL OF numerical ANALYSIS 2024年第2期44卷 886-919页

作者： Beltran, Carlos Noferini, Vanni Vannieuwenhoven, Nick Univ Cantabria Dept MATESCO Avda Los Castros S-N Santander 39005 Spain Aalto Univ Dept Math & Syst Anal POB 11100 FI-00076 Aalto Finland Katholieke Univ Leuven Dept Comp Sci Celestijnenlaan 200A B-3001 Leuven Belgium

We state some widely satisfied hypotheses, depending only on two functions g and h, under which the composition of a forward stable algorithm for g and a forward stable algorithm for h is a forward stable algorithm for the composition g ? h. We show that the failure of these conditions can potentially lead to unstable algorithms. Finally, we list a number of examples to illustrate the new concepts and the usability of the results.

关键词： Condition number numerical stability numerical algorithm composition theorem amenability relative error

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The α-path of multi-dimensional uncertain differential equations and its applications

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JOURNAL OF INTELLIGENT & FUZZY SYSTEMS 2024年第2期46卷 4761-4776页

作者： Wang, Xiaoli Wang, Chongguo Shi, Gang Xinjiang Univ Sch Comp Sci & Technol Urumqi 830046 Peoples R China

As a complex uncertain differential equation, how to solve the multi-dimensional uncertain differential equation is a complicated and difficult problem. This paper will be devoted to the a-path of some special multi-dimensional uncertain differential equations, namely, multi-factor uncertain differential equations, nested uncertain differential equations and multifactor nested uncertain differential equations. The alpha psi-path method is used to study the numerical solution problems of the above three special multi-dimensional uncertain differential equations. At the same time, the inverse uncertainty distributions and expected values of these three special multi-dimensional uncertain differential equations are also obtained. At last, the numerical algorithm examples are given to verify it.

关键词： Uncertainty theory multi-dimensional uncertain differential equation alpha-path numerical algorithm

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