Mathematics Commons^™

A Multigrid Multilevel Monte Carlo Method For Stokes–Darcy Model With Random Hydraulic Conductivity And Beavers–Joseph Condition, Zhipeng Yang, Ju Ming, Changxin Qiu, Maojun Li, Xiaoming He

Mathematics and Statistics Faculty Research & Creative Works

A multigrid multilevel Monte Carlo (MGMLMC) method is developed for the stochastic Stokes–Darcy interface model with random hydraulic conductivity both in the porous media domain and on the interface. Three interface conditions with randomness are considered on the interface between Stokes and Darcy equations, especially the Beavers–Joesph interface condition with random hydraulic conductivity. Because the randomness through the interface affects the flow in the Stokes domain, we investigate the coupled stochastic Stokes–Darcy model to improve the fidelity. Under suitable assumptions on the random coefficient, we prove the existence and uniqueness of the weak solution of the variational form. To construct …

The Beverton-Hold Model On Isolated Time Scales, Martin Bohner, Jaqueline Mesquita, Sabrina Streipert

Mathematics and Statistics Faculty Research & Creative Works

In this work, we formulate the Beverton-Holt model on isolated time scales and extend existing results known in the discrete and quantum calculus cases. Applying a recently introduced definition of periodicity for arbitrary isolated time scales, we discuss the effects of periodicity onto a population modeled by a dynamic version of the Beverton-Holt equation. The first main theorem provides conditions for the existence of a unique !-periodic solution that is globally asymptotically stable, which addresses the first Cushing-Henson conjecture on isolated time scales. The second main theorem concerns the generalization of the second Cushing-Henson conjecture. It investigates the effects of …

Optimal Equivalence Testing In Exponential Families, Renren Zhao, Robert L. Paige

Mathematics and Statistics Faculty Research & Creative Works

We develop uniformly most powerful unbiased (UMPU) two sample equivalence test for a difference of canonical parameters in exponential families. This development involves a non-unique reparameterization. We address this issue via a novel characterization of all possible reparameterizations of interest in terms of a matrix group. Furthermore, our procedure involves an intractable conditional distribution which we reproduce to a high degree of accuracy using saddle point approximations. The development of this saddle point-based procedure involves a non-unique reparameterization, but we show that our procedure is invariant under choice of reparameterization. Our real data example considers the mean-to-variance ratio for normally …

On The Hartogs Extension Theorem For Unbounded Domains In CN, Al Boggess, Roman Dwilewicz, Egmont Porten

Mathematics and Statistics Faculty Research & Creative Works

Let Ω ⊂ Cn, n > 2, be a domain with smooth connected boundary. If Ω is relatively compact, the Hartogs–Bochner theorem ensures that every CR distribution on ∂Ω has a holomorphic extension to Ω. For unbounded domains this extension property may fail, for example if Ω contains a complex hypersurface. The main result in this paper tells that the extension property holds if and only if the envelope of holomorphy of Cn \ Ω is Cn. It seems that it is the first result in the literature which gives a geometric characterization of unbounded domains in Cn for which the …

Fundamental Structure Of General Stochastic Dynamical Systems: High-Dimension Case, Haoyu Wang, Xiaoliang Gan, Wenqing Hu, Ping Ao

Mathematics and Statistics Faculty Research & Creative Works

No one has proved that mathematically general stochastic dynamical systems have a special structure. Thus, we introduce a structure of a general stochastic dynamical system. According to scientific understanding, we assert that its deterministic part can be decomposed into three significant parts: the gradient of the potential function, friction matrix and Lorenz matrix. Our previous work proved this structure for the low-dimension case. In this paper, we prove this structure for the high-dimension case. Hence, this structure of general stochastic dynamical systems is fundamental.

Asymptotic Properties Of Kneser Solutions To Third-Order Delay Differential Equations, Martin Bohner, John R. Graef, Irena Jadlovská

Mathematics and Statistics Faculty Research & Creative Works

The aim of this paper is to extend and complete the recent work by Graef et al. (J. Appl. Anal. Comput., 2021) analyzing the asymptotic properties of solutions to third-order linear delay differential equations. Most importantly, the authors tackle a particularly challenging problem of obtaining lower estimates for Kneser-type solutions. This allows improvement of existing conditions for the nonexistence of such solutions. As a result, a new criterion for oscillation of all solutions of the equation studied is established.

Maintenance Optimization In A Digital Twin For Industry 4.0, Abhijit Gosavi, Vy Khoi Le

Engineering Management and Systems Engineering Faculty Research & Creative Works

The advent of Internet of Things and artificial intelligence in the era of Industry 4.0 has transformed decision-making within production systems. In particular, many decisions that previously required significant human activity are now made automatically with minimal human intervention via so-called digital twins (DTs). In the context of maintenance and reliability modeling, this naturally calls for new paradigms that can be seamlessly integrated within DTs for decision-making. The input data for time to failure needed in reliability computations are directly collected from the work center in a digital setting and often do not satisfy a known distribution. A neural network …

Several Problems In Nonlinear Schrödinger Equations, Tim Van Hoose

Masters Theses

“We study several different problems related to nonlinear Schrödinger equations….

We prove several new results for the first equation: a modified scattering result for both an averaged version of the equation and the full equation, as well as a set of Strichartz estimates and a blowup result for the 3d cubic problem.

We also present an exposition of the classical work of Bourgain on invariant measures for the second equation in the mass-subcritical regime”--Abstract, page iv.

Oscillation Of Nonlinear Third-Order Difference Equations With Mixed Neutral Terms, Jehad Alzabut, Martin Bohner, Said R. Grace

Mathematics and Statistics Faculty Research & Creative Works

In this paper, new oscillation results for nonlinear third-order difference equations with mixed neutral terms are established. Unlike previously used techniques, which often were based on Riccati transformation and involve limsup or liminf conditions for the oscillation, the main results are obtained by means of a new approach, which is based on a comparison technique. Our new results extend, simplify, and improve existing results in the literature. Two examples with specific values of parameters are offered.

Fourth Derivative Singularly P-Stable Method For The Numerical Solution Of The Schrödinger Equation, Ali Shokri, Higinio Ramos, Mohammad Mehdizadeh Khalsaraei, Fikret A. Aliev, Martin Bohner

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we construct a method with eight steps that belongs to the family of Obrechkoff methods. Due to the explicit nature of the new method, not only does it not require another method as predictor, but it can also be considered as a suitable predictive technique to be used with implicit methods. Periodicity and error terms are studied when applied to solve the radial Schrödinger equation, considering different energy levels. We show its advantages in terms of accuracy, consistency, and convergence in comparison with other methods of the same order appearing in the literature.

Predicting Lifespan Of Drosophila Melanogaster: A Novel Application Of Convolutional Neural Networks And Zero-Inflated Autoregressive Conditional Poisson Model, Yi Zhang, V. A. Samaranayake, Gayla R. Olbricht, Matthew S. Thimgan

Mathematics and Statistics Faculty Research & Creative Works

A model to classify the lifespan of Drosophila, the fruit fly, into short- and long-lived categories based on a sleep characteristic, extracted from activity data, is developed using a two-stage process. Stage 1 models the per-minute activity counts of each fly using a zero-inflated autoregressive conditional Poisson model. These probabilities are allowed to vary hourly, reflecting the circadian and other cycles present in a fly's sleep architecture. A 5-day moving window is used to model data allowing the model parameters to vary over the course of the fly's life. The resulting probabilities capture information about changes in sleep patterns with …

Generalization Of Mitrinović–Pečarić Inequalities On Time Scales, Ahmed A. El-Deeb, Elvan Akin, Billur Kaymakçalan

Mathematics and Statistics Faculty Research & Creative Works

We prove some new inequalities of Mitrinović–Pečarić inequalities for convex functions on an arbitrary time scale using delta integrals. These inequalities extend and improve some known dynamic inequalities in the literature. The main results will be proved by using Hölder and Jensen inequalities and a simple consequence of Keller's and Poetzsche's chain rules on time scales.

Discrete Fractional Boundary Value Problems And Inequalities, Martin Bohner, Nick Fewster-Young

Mathematics and Statistics Faculty Research & Creative Works

In this paper, a general nonlinear discrete fractional boundary value problem is considered, of order between one and two. The main result is an existence theorem, proving the existence of at least one solution to the boundary value problem, subject to validity of a certain key inequality that allows unrestricted growth in the problem. The proof of this existence theorem is accomplished by using Brouwer's fixed point theorem as well as two other main results of this paper, namely, first, a result showing that the solutions of the boundary value problem are exactly the solutions to a certain equivalent integral …

Oscillation And Nonoscillation Criteria For Four-Dimensional Advanced And Delay Time-Scale Systems, Elvan Akin, Gülşah Yenpi

Mathematics and Statistics Faculty Research & Creative Works

We obtain oscillation and Non oscillation criteria for solutions to four-dimensional advanced and delay systems of first-order dynamic equations on time scales. To establish oscillation criteria, we eliminate Non oscillatory solutions of the systems based on the sign of components of the solutions. Furthermore, some of our results are new in the discrete case.

Conservative Unconditionally Stable Decoupled Numerical Schemes For The Cahn-Hilliard-Navier-Stokes-Darcy-Boussinesq System, Wenbin Chen, Daozhi Han, Xiaoming Wang, Yichao Zhang

Mathematics and Statistics Faculty Research & Creative Works

We propose two mass and heat energy conservative, unconditionally stable, decoupled numerical algorithms for solving the Cahn-Hilliard-Navier-Stokes-Darcy-Boussinesq system that models thermal convection of two-phase flows in superposed free flow and porous media. The schemes totally decouple the computation of the Cahn-Hilliard equation, the Darcy equations, the heat equation, the Navier-Stokes equations at each time step, and thus significantly reducing the computational cost. We rigorously show that the schemes are conservative and energy-law preserving. Numerical results are presented to demonstrate the accuracy and stability of the algorithms.

Dynamics Of Plane Waves In The Fractional Nonlinear Schrödinger Equation With Long-Range Dispersion, Siwei Duo, Taras I. Lakoba, Yanzhi Zhang

Mathematics and Statistics Faculty Research & Creative Works

We analytically and numerically investigate the stability and dynamics of the plane wave solutions of the fractional nonlinear Schrödinger (NLS) equation, where the long-range dispersion is described by the fractional Laplacian (−∆)α/2 . The linear stability analysis shows that plane wave solutions in the defocusing NLS are always stable if the power α ∈ [1, 2] but unstable for α ∈ (0, 1). In the focusing case, they can be linearly unstable for any α ∈ (0, 2]. We then apply the split-step Fourier spectral (SSFS) method to simulate the nonlinear stage of the plane waves dynamics. In agreement with …

Confluent Projections And Connectedness Of Inverse Limits, Włodzimierz J. Charatonik, Daria Michalik

Mathematics and Statistics Faculty Research & Creative Works

V. Nall proved that connectedness is preserved under inverse limits if the bounding functions are unions of functions with connected images. We show that for such functions the projections from the graph onto domain are confluent and we investigate relationships between functions satisfying this or similar conditions with confluence or openness of projections.

Efficient, Positive, And Energy Stable Schemes For Multi-D Poisson–Nernst–Planck Systems, Hailiang Liu, Wumaier Maimaitiyiming

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we design, analyze, and numerically validate positive and energy-dissipating schemes for solving the time-dependent multi-dimensional system of Poisson–Nernst–Planck equations, which has found much use in the modeling of biological membrane channels and semiconductor devices. The semi-implicit time discretization based on a reformulation of the system gives a well-posed elliptic system, which is shown to preserve solution positivity for arbitrary time steps. The first order (in time) fully discrete scheme is shown to preserve solution positivity and mass conservation unconditionally, and energy dissipation with only a mild O (1) time step restriction. The scheme is also shown to …

Adaptive Kriging Method For Uncertainty Quantification Of The Photoelectron Sheath And Dust Levitation On The Lunar Surface, Xinpeng Wei, Jianxun Zhao, Xiaoming He, Zhen Hu, Xiaoping Du, Daoru Han

Mechanical and Aerospace Engineering Faculty Research & Creative Works

This paper presents an adaptive Kriging based method to perform uncertainty quantification (UQ) of the photoelectron sheath and dust levitation on the lunar surface. The objective of this study is to identify the upper and lower bounds of the electric potential and that of dust levitation height, given the intervals of model parameters in the one-dimensional (1D) photoelectron sheath model. To improve the calculation efficiency, we employ the widely used adaptive Kriging method (AKM). A task-oriented learning function and a stopping criterion are developed to train the Kriging model and customize the AKM. Experiment analysis shows that the proposed AKM …

Photoelectron Sheath Near The Lunar Surface: Fully Kinetic Modeling And Uncertainty Quantification Analysis, Jianxun Zhao, Xinpeng Wei, Zhangli Hu, Xiaoming He, Daoru Frank Han, Zhen Hu, Xiaoping Du

Mathematics and Statistics Faculty Research & Creative Works

This paper presents a modeling and uncertainty quantification (UQ) study of the photoelectron sheath near the lunar surface. A fully kinetic 3-D finite-difference (FD) particle-in-cell (PIC) code is utilized to simulate the plasma interaction near the lunar surface and the resulting photoelectron sheath. For the uncertainty quantification analysis, this FD-PIC code is treated as a black box providing high-fidelity quantities of interest, which are also used to construct efficient reduced-order models to perform UQ analysis. 1-D configuration is chosen to present the analytic sheath solution as well as to demonstrate the procedure and capability of the UQ analysis.

Fully-Kinetic Particle-In-Cell Simulations Of Photoelectron Sheath On Uneven Lunar Surface, Jianxun Zhao, Xinpeng Wei, Xiaoming He, Daoru Frank Han, Xiaoping Du

Mathematics and Statistics Faculty Research & Creative Works

This paper presents a modeling and simulation study of the photoelectron sheath near uneven lunar surface. A fully kinetic 3-D finite-difference (FD) particle-in-cell (PIC) code is utilized to simulate the plasma interaction with local uneven surface terrain on the lunar surface in 2-D photoelectron sheaths. The code is first validated using a 1-D plasma charging and sheath problem by comparing with a semi-analytic solution. Good agreement is obtained. The 2-D FD-PIC simulations present the distributions of electric potential and charged species densities near the uneven lunar surface. It shows that the surface potential is highly influenced by the exposure to …

Stokes-Darcy System, Small-Darcy-Number Behaviour And Related Interfacial Conditions, Wenqi Lyu, Xiaoming Wang

Mathematics and Statistics Faculty Research & Creative Works

We show that the Stokes-Darcy system, which governs flows through adjacent porous and pure-fluid domains in the two-domain approach without forced filtration, can be recovered from the Helmholtz minimal dissipation principle. While the continuity of normal velocity across the interface is imposed explicitly for mass conservation, only the Beavers-Joseph-Saffman-Jones (BJSJ) interface boundary condition is imposed implicitly, and the balance of the normal-force interface boundary condition appears naturally in the variational process. This set of interface boundary conditions is well-accepted in the mathematics community. We show that these interfacial boundary conditions, at the physically important small-Darcy-number regime, are consistent with continuity …

A Deep Learning Model To Predict Traumatic Brain Injury Severity And Outcome From Mr Images, Dacosta Yeboah, Hung Nguyen, Daniel B. Hier, Gayla R. Olbricht, Tayo Obafemi-Ajayi

Chemistry Faculty Research & Creative Works

For Many Neurological Disorders, Including Traumatic Brain Injury (TBI), Neuroimaging Information Plays a Crucial Role Determining Diagnosis and Prognosis. TBI is a Heterogeneous Disorder that Can Result in Lasting Physical, Emotional and Cognitive Impairments. Magnetic Resonance Imaging (MRI) is a Non-Invasive Technique that Uses Radio Waves to Reveal Fine Details of Brain Anatomy and Pathology. Although MRIs Are Interpreted by Radiologists, Advances Are Being Made in the Use of Deep Learning for MRI Interpretation. This Work Evaluates a Deep Learning Model based on a Residual Learning Convolutional Neural Network that Predicts TBI Severity from MR Images. the Model Achieved a …

New Proper Orthogonal Decomposition Approximation Theory For Pde Solution Data, Sarah Locke, John R. Singler

Mathematics and Statistics Faculty Research & Creative Works

In our previous work [J. R. Singler, SIAM J. Numer. Anal., 52 (2014), pp. 852- 876], we considered the proper orthogonal decomposition (POD) of time varying PDE solution data taking values in two different Hilbert spaces. We considered various POD projections of the data and obtained new results concerning POD projection errors and error bounds for POD reduced order models of PDEs. In this work, we improve on our earlier results concerning POD projections by extending to a more general framework that allows for nonorthogonal POD projections and seminorms. We obtain new exact error formulas and convergence results for POD …

A Natural Frenet Frame For Null Curves On The Lightlike Cone In Minkowski Space ℝ⁴₂, Nemat Abazari, Martin Bohner, Ilgin Sağer, Alireza Sedaghatdoost, Yusuf Yayli

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we investigate the representation of curves on the lightlike cone ℚ³₂ in Minkowski space ℝ⁴₂ by structure functions. In addition, with this representation, we classify all of the null curves on the lightlike cone ℚ³₂ in four types, and we obtain a natural Frenet frame for these null curves. Furthermore, for this natural Frenet frame, we calculate curvature functions of a null curve, especially the curvature function κ₂ = 0 , and we show that any null curve on the lightlike cone is a helix. Finally, we find all curves with constant curvature functions.

Hereditarily Irreducible Maps, Hussam Abobaker, Włodzimierz J. Charatonik

Mathematics and Statistics Faculty Research & Creative Works

A map f:X→Y from a continuum X onto a continuum Y is said to be hereditarily irreducible, if f(A)⊊f(B) for any subcontinua A and B such that A⊊B. We investigate properties of hereditarily irreducible maps between continua. Special attention is given to maps between graphs and maps from the interval.

Inverse Limits And Atomic Projections, Włodzimierz J. Charatonik, Faruq A. Mena, Robert Paul Roe

Mathematics and Statistics Faculty Research & Creative Works

We consider generalized inverse limits of continua with bonding functions Fn that have the projection of Graph (Fn) onto the second (first) factor atomic and images (pre-image) of points are zero-dimensional. For such bonding functions we show that under some easily verified conditions that if the first (all) factor space(s) has a certain property then the inverse limit space must have this property. The properties considered include hereditary decomposability, hereditary indecomposability, hereditary unicoherence, arc-likeness, and tree-likeness. We illustrate the theorems by several examples.

Energy Stable Numerical Schemes For Ternary Cahn-Hilliard System, Wenbin Chen, Cheng Wang, Shufen Wang, Xiaoming Wang, Steven M. Wise

Mathematics and Statistics Faculty Research & Creative Works

We present and analyze a uniquely solvable and unconditionally energy stable numerical scheme for the ternary Cahn-Hilliard system, with a polynomial pattern nonlinear free energy expansion. One key difficulty is associated with presence of the three mass components, though a total mass constraint reduces this to two components. Another numerical challenge is to ensure the energy stability for the nonlinear energy functional in the mixed product form, which turns out to be non-convex, non-concave in the three-phase space. to overcome this subtle difficulty, we add a few auxiliary terms to make the combined energy functional convex in the three-phase space, …

On The Noisy Gradient Descent That Generalizes As Sgd, Jingfeng Wu, Wenqing Hu, Haoyi Xiong, Jun Huan, Vladimir Braverman, Zhanxing Zhu

Mathematics and Statistics Faculty Research & Creative Works

The gradient noise of SGD is considered to play a central role in the observed strong generalization abilities of deep learning. While past studies confirm that the magnitude and covariance structure of gradient noise are critical for regularization, it remains unclear whether or not the class of noise distributions is important. In this work we provide negative results by showing that noises in classes different from the SGD noise can also effectively regularize gradient descent. Our finding is based on a novel observation on the structure of the SGD noise: it is the multiplication of the gradient matrix and a …

Vanishing Porosity Limit Of The Coupled Stokes-Brinkman System, Mingwen Fei, Dongjuan Niu, Xiaoming Wang

Mathematics and Statistics Faculty Research & Creative Works

We investigate the small porosity asymptotic behavior of the coupled Stokes-Brinkman system in the presence of a curved interface between the Stokes region and the Brinkman region. in particular, we derive a set of approximate solutions, validated via rigorous analysis, to the coupled Stokes-Brinkman system. of particular interest is that the approximate solution satisfies a generalized Beavers-Joseph-Saffman-Jones interface condition (1.9) with the constant of proportionality independent of the curvature of the interface.