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Mathematics and Statistics Faculty Research & Creative Works

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Full-Text Articles in Mathematics

Revisiting Ulam Stability For Boundary Value Problems, Martin Bohner, Snezhana Hristova, Agnieszka B. Malinowska, Ewa Girejko Dec 2026

Revisiting Ulam Stability For Boundary Value Problems, Martin Bohner, Snezhana Hristova, Agnieszka B. Malinowska, Ewa Girejko

Mathematics and Statistics Faculty Research & Creative Works

The main goal of this paper is to apply Ulam stability theory to boundary value problems for dynamic equations, while addressing several common misconceptions found in the existing literature. We identify the key issues that arise when applying Ulam stability to such problems and propose three distinct approaches to overcome them. To enhance clarity and accessibility, we begin with nonlinear ordinary differential equations and subsequently extend the analysis to nonlinear dynamic equations on time scales. Since a time scale is defined as any nonempty closed subset of the real numbers, our results are applicable to dynamic equations on continuous, discrete, …

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A Stabilized Weighted Interior Penalty Method For Thermal Convection Model In Heterogeneous Porous Media, Yuanyuan Hou, Qianqian Ding, Xiaoming He, Yanping Lin Aug 2026

A Stabilized Weighted Interior Penalty Method For Thermal Convection Model In Heterogeneous Porous Media, Yuanyuan Hou, Qianqian Ding, Xiaoming He, Yanping Lin

Mathematics and Statistics Faculty Research & Creative Works

In this article, we propose and analyze a stabilized weighted interior penalty method for solving the thermal convection problems in heterogeneous porous media. We first transform the thermal convection model into the pressure primal form with homogeneous Neumann boundary condition and develop a symmetric weighted interior penalty method to handle the discontinuous Darcy number and automatically adjust the penalty coefficient corresponding to the varying permeability. Then we recover the velocity by a stabilized method and incorporate it into the energy equation to obtain the temperature. The stability and convergence rates of the numerical solutions are rigorously proved and verified by …

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Phase Field Modeling And A Fully Discrete Numerical Scheme For Two-Phase Incompressible Mhd Flows With Different Densities, Electric Conductivities And Viscosities, Xiaoyong Chen, Rui Li, Jian Li, Xiaoming He, Yanping Lin May 2026

Phase Field Modeling And A Fully Discrete Numerical Scheme For Two-Phase Incompressible Mhd Flows With Different Densities, Electric Conductivities And Viscosities, Xiaoyong Chen, Rui Li, Jian Li, Xiaoming He, Yanping Lin

Mathematics and Statistics Faculty Research & Creative Works

This article establishes a phase field model for governing the two-phase incompressible MHD flows with different densities, electric conductivities, and viscosities. In addition to the coupling between the Cahn–Hilliard phase field equations and the single-phase MHD equations together with the varying parameters, it is physically faithful and mathematically rigorous for the modeling to incorporate a relative flux term, which is related to the diffusion of the components, into the coupled system, inspired by Abels et al. and Shen and Yang. We present a linear fully discrete numerical scheme for this complex multi-physics system, which leverages the artificial compressibility method, an …

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A Low-Rank Solver For The Stokes–Darcy Model With Random Hydraulic Conductivity And Beavers–Joseph Condition, Yujun Zhu, Yulan Ning, Zhipeng Yang, Xiaoming He, Ju Ming May 2026

A Low-Rank Solver For The Stokes–Darcy Model With Random Hydraulic Conductivity And Beavers–Joseph Condition, Yujun Zhu, Yulan Ning, Zhipeng Yang, Xiaoming He, Ju Ming

Mathematics and Statistics Faculty Research & Creative Works

This paper proposes, analyzes, and demonstrates an efficient low-rank solver for the stochastic Stokes-Darcy interface model with a random hydraulic conductivity both in the porous media domain and on the interface. We consider three interface conditions with randomness, including the Beavers–Joseph interface condition with the random hydraulic conductivity, on the interface between the free flow and the porous media flow. Our solver employs a novel generalized low-rank approximation of the large-scale stiffness matrices, which can significantly cut down the computational costs and memory requirements associated with matrix inversion without losing accuracy. Therefore, by adopting a suitable data compression ratio, the …

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The Kaczmarz Algorithm In Hilbert C∗-Modules, Daniel Alpay, Chad Berner, Eric S. Weber Apr 2026

The Kaczmarz Algorithm In Hilbert C∗-Modules, Daniel Alpay, Chad Berner, Eric S. Weber

Mathematics and Statistics Faculty Research & Creative Works

The Kaczmarz algorithm in Hilbert spaces is a classical iterative method for stably recovering vectors from inner product data. In this paper, we extend the algorithm to the setting of Hilbert C∗-modules and establish analogues of its effectiveness in both finite-dimensional and stationary cases. Consequently, we demonstrate that continuous families of elements in a Hilbert space can be uniformly recovered using the Kaczmarz algorithm. Additionally, we develop a normalized Cauchy transform for continuous families of measures and use it to provide sufficient conditions under which standard frames in Hilbert C(X)-modules can be generated by the Kaczmarz algorithm and …

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A Fully Discrete Decoupled Scheme And Applications For Non-Isothermal Two-Phase Flow Model With Different Viscosities And Thermal Diffusivities, Jian Li, Chunhao Chen, Xiaoyong Chen, Rui Li, Xiaoming He Apr 2026

A Fully Discrete Decoupled Scheme And Applications For Non-Isothermal Two-Phase Flow Model With Different Viscosities And Thermal Diffusivities, Jian Li, Chunhao Chen, Xiaoyong Chen, Rui Li, Xiaoming He

Mathematics and Statistics Faculty Research & Creative Works

To study natural convection problems in two-phase flows, a non-isothermal two-phase flow model incorporating differential viscosities and thermal diffusivities is considered and analyzed via the phase-field method. This modeling framework involves the Multiphysics coupling of the Cahn-Hilliard phase field equations, heat transfer equation, and Navier-Stokes equations, resulting in a strongly nonlinear system. To efficiently solve the sophisticated system, we develop, analyze, and demonstrate a decoupled linear fully discrete scheme, which leverages the invariant energy quadratization strategy for the Cahn-Hilliard phase field system, the artificial compressibility method without artificial pressure boundary condition, an explicit-implicit treatment of nonlinear terms, and the addition …

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Sequences That Do Frame Reconstruction, Chad Berner Apr 2026

Sequences That Do Frame Reconstruction, Chad Berner

Mathematics and Statistics Faculty Research & Creative Works

Frames allow all elements of a Hilbert space to be reconstructed by inner product data in a stable manner. Recently, there is interest in relaxing the definition of frames to understand the implications for stable signal recovery. In this paper, we relax the definition of a frame by allowing the operator in the frame decomposition formula to not be invertible. We provide a complete classification of sequences that allow this decomposition via a type of frame operator. In addition, we provide several examples of sequences that allow this reconstruction property that are not frames and illustrate in which ways they …

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Averaging Principle For A General Class Of Periodic Functions In Discrete Spaces, Martin Bohner, Jaqueline G. Mesquita, Sabrina Streipert Mar 2026

Averaging Principle For A General Class Of Periodic Functions In Discrete Spaces, Martin Bohner, Jaqueline G. Mesquita, Sabrina Streipert

Mathematics and Statistics Faculty Research & Creative Works

In this work, we develop a periodic averaging principle for arbitrary discrete time domains, leveraging a novel definition of periodicity. This definition does not rely on the classical requirement for the time domain itself to be periodic. We implement this averaging principle across diverse discrete time domains and explore a range of periodic functions within this extended context. The paper contains several examples with numerical simulations, providing visual demonstrations of our results. This highlights the versatility of our averaging principle and its potential to understand dynamics of nonautonomous recurrences with complex temporal patterns.

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Controllability Of The Semilinear Benjamin–Bona–Mahony Dynamic Equation On Homogeneous Time Scales, Martin Bohner, Cosme Duque, Hugo Leiva Mar 2026

Controllability Of The Semilinear Benjamin–Bona–Mahony Dynamic Equation On Homogeneous Time Scales, Martin Bohner, Cosme Duque, Hugo Leiva

Mathematics and Statistics Faculty Research & Creative Works

This work investigates the approximate controllability and free-time approximate controllability of a generalized semi linear Benjamin–Bona–Mahony type dynamic equation defined on homogeneous time scales, subject to homogeneous Dirichlet boundary conditions. To accomplish this, the problem is framed within an abstract setting, employing the -semigroup theory on time scales. Moreover, we apply a technique introduced by Bashirov et al. [1, 2], which enables us to avoid relying on fixed point theorems.

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A Diffuse Interface Model And Fully Decoupled, Energy-Stable Scheme For The Two-Phase Ferrofluid Flows In Porous Media, Guo Dong Zhang, Shuai Zhou, Yunqing Huang, Xiaoming He, Xiaofeng Yang Mar 2026

A Diffuse Interface Model And Fully Decoupled, Energy-Stable Scheme For The Two-Phase Ferrofluid Flows In Porous Media, Guo Dong Zhang, Shuai Zhou, Yunqing Huang, Xiaoming He, Xiaofeng Yang

Mathematics and Statistics Faculty Research & Creative Works

We propose a new diffuse interface model for two-phase porous media ferrofluid flows, employing the phase-field method and establishing an associated energy law. This thermodynamically consistent multi-physics model integrates the Cahn-Hilliard equations, Darcy equations, the magnetostatic equation, and magnetization equations. To efficiently solve the system by addressing its inherent nonlinearities, coupling, and saddle-point structure, we incorporate several advanced techniques, including the SAV-ZEC method to handle nonlinearities and couplings, a reformulated approach to decouple the linear coupling of the magnetic potential and magnetization, and the pressure projection method to decouple velocity and pressure. This results in a numerical scheme that is …

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An Seir Model On Time Scales With Discrete Applications To Tuberculosis, E. Akın, G. Yeni, D. Konur, S. R. Işık, M. R. Işık Mar 2026

An Seir Model On Time Scales With Discrete Applications To Tuberculosis, E. Akın, G. Yeni, D. Konur, S. R. Işık, M. R. Işık

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we propose a novel dynamical model on time scales consisting of new parameters to investigate the transmission dynamics of tuberculosis (TB), one of the deadliest infectious diseases worldwide, characterized by a long latency stage. The dynamical TB model, governed by the Susceptible–Exposed–Infected–Recovered (SEIR) framework within a unified form, yields a continuous model with a non-saturated incidence rate on the real numbers and discrete models with saturated incidence rates when different time domains are chosen. We analyze the stability of the equilibrium points of both the continuous TB model on the set of real numbers and the discrete …

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Pointwise Self-Homeomorphic Generalized Inverse Limits, Ali H. Ali, Faruq A. Mena, Robert Paul Roe Mar 2026

Pointwise Self-Homeomorphic Generalized Inverse Limits, Ali H. Ali, Faruq A. Mena, Robert Paul Roe

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we find uncountable families of generalized inverse sequences on intervals, where the bonding functions consist of a finite number of line segments, such that the inverse limit spaces of these sequences are pointwise self-homeomorphic continua. We give several examples of pointwise self-homeomorphic continua obtained in this manner including the dendrite D3 and a dendrite containing Dω. The dendrite D3 was obtained previously, by others, as a generalized inverse limit but the bonding function in that example contained infinitely many line segments. We show that the techniques we use on intervals can be extended to inverse limits where …

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Positive Solutions Of Semipositone Singular Three-Points Boundary Value Problems For Nonlinear Fractional Differential Equations, Xueyan Zhang, Zhaocai Hao, Martin Bohner Feb 2026

Positive Solutions Of Semipositone Singular Three-Points Boundary Value Problems For Nonlinear Fractional Differential Equations, Xueyan Zhang, Zhaocai Hao, Martin Bohner

Mathematics and Statistics Faculty Research & Creative Works

This study introduces the existence of one positive solution for a specific category of semipositive singular three-point boundary value problems associated with Caputo fractional differential equations. The proof relies on the application of the Guo–Krasnosel'skii fixed point theorem. In the end, we provide an illustrative example.

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Meshless Collocation Methods For Time-Dependent Nonlocal Problems Based On Radial Basis Functions, Qiao Zhuang, Yanzhi Zhang, Zhongqiang Zhang Jan 2026

Meshless Collocation Methods For Time-Dependent Nonlocal Problems Based On Radial Basis Functions, Qiao Zhuang, Yanzhi Zhang, Zhongqiang Zhang

Mathematics and Statistics Faculty Research & Creative Works

We present radial basis function (RBF) collocation methods for time-dependent space fractional problems on general bounded domains. Building on a recently developed approach for accurately computing the integral fractional Laplacian of any RBF, we design collocation schemes for fractional heat and Stokes equations using extended-domain techniques. In particular, we propose a numerical Leray projection method for fractional Stokes problems, where both the discrete projection operator and the collocation scheme are formulated on extended domains to handle complex domains. Numerical results demonstrate the effectiveness of the proposed methods in solving time-dependent nonlocal problems on complex domains.

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Fourier Pseudospectral Methods For The Variable-Order Space Fractional Wave Equations, Yanzhi Zhang, Xiaofei Zhao, Shiping Zhou Jan 2026

Fourier Pseudospectral Methods For The Variable-Order Space Fractional Wave Equations, Yanzhi Zhang, Xiaofei Zhao, Shiping Zhou

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we propose Fourier pseudospectral methods to solve the variable-order space fractional wave equation and develop an accelerated matrix-free approach for its effective implementation. In constant-order cases, fast algorithms can be designed via the fast Fourier transforms (FFTs), and the computational cost at each time step is O(Nlog⁡N) with N the total number of spatial points. In variable-order cases, however, the spatial dependence in the power s(x) leads to the failure of inverse FFTs. While the direct matrix-vector multiplication approach becomes impractical due to excessive memory requirements. Hence, we propose an accelerated matrix-free approach for effective implementation in …

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Partially Penalized Anisotropic Trilinear Ife-Pic Methods For Dc Plasma Transport Problems, Jiahui Li, Guangqing Xia, Yajie Han, Ziping Wang, Chang Lu, Xiaoming He Jan 2026

Partially Penalized Anisotropic Trilinear Ife-Pic Methods For Dc Plasma Transport Problems, Jiahui Li, Guangqing Xia, Yajie Han, Ziping Wang, Chang Lu, Xiaoming He

Mathematics and Statistics Faculty Research & Creative Works

Implicit and hybrid particle-in-cell methods are widely used for efficient simulation of DC discharge plasma transport. However, their computations require solving anisotropic elliptic equations and face challenges related to mesh geometry, non-axisymmetry, and complex interfaces. Moreover, the accuracy of particle trajectories is critical for plasma etching and erosion studies, where errors near interfaces can significantly impact simulation results. To address these challenges, this paper proposes a three-dimensional anisotropic trilinear partially penalized immersed finite element (ATPPIFE) method, which captures interfaces on Cartesian meshes and effectively reduces discontinuities at interface element faces, ensuring that particle trajectories better align with real-world behavior. Building …

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Special Issue: Innovative Numerical Approaches For Problems In Science And Engineering, Xiaoming He, Shuhao Cao, Qiao Zhuang Jan 2026

Special Issue: Innovative Numerical Approaches For Problems In Science And Engineering, Xiaoming He, Shuhao Cao, Qiao Zhuang

Mathematics and Statistics Faculty Research & Creative Works

No abstract provided.

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Equilibrium Stability Under Nuclear Confrontation, Martin Bohner, A. A. Martynyuk Jan 2026

Equilibrium Stability Under Nuclear Confrontation, Martin Bohner, A. A. Martynyuk

Mathematics and Statistics Faculty Research & Creative Works

This article proposes and analyzes mathematical models of confrontation between two and n countries, including countries with nuclear weapons. The proposed models are based on a generalization of Richardson's well-known mathematical model of the arms race. Namely, the factor of hostility is filled with expanded content, including public opinion and the armed forces of the opposing countries. Qualitative analysis of confrontation models is carried out by the method of Lyapunov functions and by applying nonlinear integral inequalities. As a result of the analysis, the conditions for the stability of the equilibrium state of the opposing countries are established, and the …

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Threshold Asymmetric Conditional Autoregressive Range (Tacarr) Model, Isuru Ratnayake, V. A. Samaranayake Jan 2026

Threshold Asymmetric Conditional Autoregressive Range (Tacarr) Model, Isuru Ratnayake, V. A. Samaranayake

Mathematics and Statistics Faculty Research & Creative Works

This paper introduces a Threshold Asymmetric Conditional Autoregressive Range (TACARR) model for analyzing the daily price ranges of financial assets. The proposed formulation assumes that the conditional expected range switches between two regimes, representing upward and downward market states, with the disturbance distribution also allowed to vary across regimes. A self-adjusting threshold component, determined by past values of the series, is used to identify the prevailing market regime. In this way, the model is able to capture asymmetric and heteroscedastic volatility behavior in financial markets. The TACARR model is designed to address several limitations of existing price range models, including …

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Infimum Dimension Nash Embeddings For 2d Projective Shape Analysis, Robert L. Paige, Vic Patrangenaru Jan 2026

Infimum Dimension Nash Embeddings For 2d Projective Shape Analysis, Robert L. Paige, Vic Patrangenaru

Mathematics and Statistics Faculty Research & Creative Works

Vector embeddings make complicated data extracted from networks, words and images, more amendable to data science applications. At the present time, the Veronese-Whitney (VW) matrix embedding of the real projective space is the state of the art for making inference about digital images from an uncalibrated camera, such as a cell phone or security camera. In this work we consider vector embeddings for the projective shape data and in particular determine the minimum dimension isometric (distance-preserving or Nash) vector embedding for a projective space. We determine such an embedding for the projective plane in closed-form. From this embedding we determine …

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A Fully Discrete Semi-Implicit Numerical Scheme And Its Optimal Error Estimates For Cahn-Hilliard-Mhd Model With Variable Density, Dongmei Duan, Fuzheng Gao, Xiaoming He, Yanping Lin Jan 2026

A Fully Discrete Semi-Implicit Numerical Scheme And Its Optimal Error Estimates For Cahn-Hilliard-Mhd Model With Variable Density, Dongmei Duan, Fuzheng Gao, Xiaoming He, Yanping Lin

Mathematics and Statistics Faculty Research & Creative Works

This paper proposes and analyzes a fully discrete semi-implicit unconditionally energy stable numerical scheme to solve the Cahn-Hilliard Magnetohydrodynamics (Cahn-Hilliard-MHD) model with variable density. The unconditional energy stability and optimal L2 error estimates are established for the fully discrete scheme. Major challenges in error estimation arise from the variable density, the strong nonlinearities, and the multi-physics coupling of the model. Under the mathematical induction framework, the Ritz quasi-projection and the Stokes quasi-projection, proposed in [SIAM J. Numer. Anal., 61(3):1218-1245, 2023], are utilized to avoid the gradient terms of the projection errors. The H−1 superconvergence error estimates of Ritz …

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Wacsaw: An Adaptive, Statistical Method To Classify Movement Into Sleep And Wakefulness States, Austin Vandegriffe, V. A. Samaranayake, Matthew S. Thimgan Dec 2025

Wacsaw: An Adaptive, Statistical Method To Classify Movement Into Sleep And Wakefulness States, Austin Vandegriffe, V. A. Samaranayake, Matthew S. Thimgan

Mathematics and Statistics Faculty Research & Creative Works

Wearable actimeters can improve our understanding of sleep in the natural environments. Current algorithms may produce inaccuracies in specific individuals and circumstances, such as quiet wakefulness. New hardware allows data collection at higher frequencies enabling sophisticated analytical methods. We have developed a novel statistical algorithm, the Wasserstein Algorithm for Classifying Sleep and Wakefulness (WACSAW), to identify behavioral states from recordings of everyday movement. WACSAW employs optimal transport techniques to identify segments with differing activity variability. Functions characterizing the segments' movement distributions were clustered into two groups using a k-nearest neighbors and labeled as sleep or wake based on their proximity …

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Generalized Transversality Conditions For Fuzzy Quantum-Symmetric Variational Problems Via Granular Approach, Martin Bohner, Ewa Girejko, Agnieszka B. Malinowska, Linh Nguyen, Baruch Schneider, Tri Truong Sep 2025

Generalized Transversality Conditions For Fuzzy Quantum-Symmetric Variational Problems Via Granular Approach, Martin Bohner, Ewa Girejko, Agnieszka B. Malinowska, Linh Nguyen, Baruch Schneider, Tri Truong

Mathematics and Statistics Faculty Research & Creative Works

This paper investigates fuzzy q-symmetric variational problems with natural boundary conditions. Based on the relative distance measure fuzzy arithmetic and horizontal membership functions (HMFs), we propose novel concepts of differentiability and integrability for fuzzy functions on quantum geometric subsets of real numbers. Then, fundamental foundations of q-symmetric calculus of variations based on HMFs are provided. With the help of HMFs and granular q-symmetric differentiability, we derive necessary optimality conditions for fuzzy q-symmetric variational problems that depend on free endpoints. Moreover, sufficient conditions for minimizers of q-symmetric variational problems are obtained. Some numerical examples illustrating the proposed approach are presented.

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Novel Statistical And Topological Data Analyses Of 2d Electronic Images, Robert L. Paige, Vic Patrangenaru Sep 2025

Novel Statistical And Topological Data Analyses Of 2d Electronic Images, Robert L. Paige, Vic Patrangenaru

Mathematics and Statistics Faculty Research & Creative Works

In this paper, novel statistical and topological data analyses of 2D images are developed. One considers methodologies based on the Region Covariance Descriptor (RCD) and Topological Data Analysis (TDA) rooted in the simplicial as well as cubical persistent homologies. These methods provide statistical methods for data from populations of complex data objects that are elements of non-Euclidean spaces. The 2D image data considered consist of pictures of two leaves—A and B—from the same tree, twenty of each leaf, from different perspectives. The novel statistical procedures developed are used for correctly determining that leaf A images and leaf B images are …

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Energy-Stable And Efficient Finite Element Schemes For The Shliomis Model Of Ferrofluid Flows, Guo Dong Zhang, Kejia Pan, Xiaoming He, Xiaofeng Yang Aug 2025

Energy-Stable And Efficient Finite Element Schemes For The Shliomis Model Of Ferrofluid Flows, Guo Dong Zhang, Kejia Pan, Xiaoming He, Xiaofeng Yang

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we aim to design two energy-stable and efficient finite element schemes for simulating the ferrofluid flows based on the well-known Shliomis model. The model is a highly nonlinear, coupled, multi-physics system, consisting of the Navier–Stokes equations, magnetostatic equation, and magnetization field equation. We propose two reliable numerical algorithms with the following desired features: linearity and unconditional energy stability. Several key techniques are used to achieve the required features, including the auxiliary variable method, consistent terms method, prediction-correction method, and semi-implicit stabilization method. The first scheme is based on a hybrid continuous/discontinuous finite elements spatial approximation, and the …

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Ceno: Non-Uniform, Segment And Parallel Zero-Knowledge Virtual Machine, Tianyi Liu, Zhenfei Zhang, Yuncong Zhang, Wenqing Hu, Ye Zhang Jun 2025

Ceno: Non-Uniform, Segment And Parallel Zero-Knowledge Virtual Machine, Tianyi Liu, Zhenfei Zhang, Yuncong Zhang, Wenqing Hu, Ye Zhang

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we explore a novel Zero-knowledge Virtual Machine (zkVM) framework leveraging succinct, non-interactive zero-knowledge proofs for verifiable computation over any code. Our approach divides the proof of program execution into two stages. In the first stage, the process breaks down program execution into segments, identifying and grouping identical sections. These segments are then proved through data-parallel circuits that allow for varying amounts of duplication. In the subsequent stage, the verifier examines these segment proofs, reconstructing the program's control and data flow based on the segments' duplication number and the original program. The second stage can be further attested …

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A Selective Discontinuous Galerkin Implicit Particle-In-Cell Method For Plasma Simulation With Improved Interpolation, Siyu Wu, Yang Li, Hongtao Liu, Xiaoming He, Yong Cao May 2025

A Selective Discontinuous Galerkin Implicit Particle-In-Cell Method For Plasma Simulation With Improved Interpolation, Siyu Wu, Yang Li, Hongtao Liu, Xiaoming He, Yong Cao

Mathematics and Statistics Faculty Research & Creative Works

This article dynamically incorporates multiple ideas into the existing direct implicit particle-in-cell (DIPIC) method for plasma simulation, in order to dramatically improve the DIPIC method for its local mesh refinement needs based on Cartesian meshes as well as its interpolation needs based on the locally refined meshes. One key tool is to utilize the selective discontinuous Galerkin method, which is based on the interior penalty discontinuous Galerkin formulation and the regular local finite element basis functions, as the electric field solver in the DIPIC simulation. This hybrid type finite element method combines the advantages of both continuous and discontinuous finite …

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Some New Hardy-Type Inequalities With Negative Parameters On Time Scales, Martin Bohner, Irena Jadlovská, Ahmed I. Saied Apr 2025

Some New Hardy-Type Inequalities With Negative Parameters On Time Scales, Martin Bohner, Irena Jadlovská, Ahmed I. Saied

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we present new Hardy-type inequalities with negative parameters on a time scale T. The adopted approach draws upon the use of a reversed Hölder dynamic inequality, a chain rule, and the integration by parts rule on time scales. In the continuous case, our results contain integral inequalities due to Benaissa and Budak, while in the discrete case, the obtained inequalities are essentially new. Additionally, we demonstrate the applicability of our results in the quantum case.

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Fourier Series For Singular Measures In Higher Dimensions, Chad Berner, John E. Herr, Palle E.T. Jorgensen, Eric S. Weber Feb 2025

Fourier Series For Singular Measures In Higher Dimensions, Chad Berner, John E. Herr, Palle E.T. Jorgensen, Eric S. Weber

Mathematics and Statistics Faculty Research & Creative Works

For multi-variable finite measure spaces, we present in this paper a new framework for non-orthogonal L2 Fourier expansions. Our results hold for probability measures μ with finite support in Rd that satisfy a certain disintegration condition that we refer to as "slice-singular". In this general framework, we present explicit L2(μ)-Fourier expansions, with Fourier exponentials having positive Fourier frequencies in each of the d coordinates. Our Fourier representations apply to every f∈L2(μ), are based on an extended Kaczmarz algorithm, and use a new recursive μ Rokhlin disintegration representation. In detail, our Fourier series expansion for f is in terms of the …

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Unconditionally Optimal Convergent Zero-Energy-Contribution Scheme For Two Phase Mhd Model, Jinjin Yang, Shipeng Mao, Xiaoming He Feb 2025

Unconditionally Optimal Convergent Zero-Energy-Contribution Scheme For Two Phase Mhd Model, Jinjin Yang, Shipeng Mao, Xiaoming He

Mathematics and Statistics Faculty Research & Creative Works

This paper focuses on the unconditionally optimal error estimates of a fully discrete decoupled scheme for two-phase magnetohydrodynamic (MHD) model with different viscosities and electric conductivities, by using the zero-energy-contribution (ZEC) method for the temporal discretization and mixed finite elements for the spatial discretization. Based on the ZEC property of the nonlinear and coupled terms of the model, an ordinary differential equation is designed to introduce a nonlocal scalar auxiliary variable which will play a key role in the design and the energy stability of the decoupled scheme. Combining fully explicit treatment on the nonlinear and coupled terms with the …

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