Mathematics Commons^™

Modeling And An Immersed Finite Element Method For An Interface Wave Equation, Jinwei Bai, Yong Cao, Xiaoming He, Hongyan Liu, Xiaofeng Yang

Mathematics and Statistics Faculty Research & Creative Works

The electromagnetic field, which is governed by Maxwell's equation, plays a key role in plasma simulation. In this article, we first derive the interface conditions when we rewrite the interface Maxwell's equation, whose problem domain involves complex media such as objects of different materials, into a parabolic–hyperbolic type of interface model, which is a modified wave equation by adding a first order time derivative term due to the lossy medium. Based on the interface conditions and the existing bilinear immersed finite element space for the interface Poisson equation, we propose an immersed finite element method for the spatial discretization of …

A Further Extension Of The Extended Riemann-Liouville Fractional Derivative Operator, Martin Bohner, Gauhar Rahman, Shahid Mubeen, Kottakkaran Sooppy Nisar

Mathematics and Statistics Faculty Research & Creative Works

The main objective of this paper is to establish the extension of an extended fractional derivative operator by using an extended beta function recently defined by Parmar et al. by considering the Bessel functions in its kernel. We also give some results related to the newly defined fractional operator, such as Mellin transform and relations to extended hypergeometric and Appell's function via generating functions.

Dynamics Of Paramagnetic And Ferromagnetic Ellipsoidal Particles In Shear Flow Under A Uniform Magnetic Field, Christopher A. Sobecki, Jie Zhang, Yanzhi Zhang, Cheng Wang

Mathematics and Statistics Faculty Research & Creative Works

We investigate the two-dimensional dynamic motion of magnetic particles of ellipsoidal shapes in shear flow under the influence of a uniform magnetic field. In the first part, we present a theoretical analysis of the rotational dynamics of the particles in simple shear flow. By considering paramagnetic and ferromagnetic particles, we study the effects of the direction and strength of the magnetic field on the particle rotation. The critical magnetic-field strength, at which particle rotation is impeded, is determined. In a weak-field regime (i.e., below the critical strength) where the particles execute complete rotations, the symmetry property of the rotational velocity …

A Multi-Step Nonlinear Dimension-Reduction Approach With Applications To Bigdata, R. Krishnan, V. A. Samaranayake, Jagannathan Sarangapani

Mathematics and Statistics Faculty Research & Creative Works

In this paper, a multi-step dimension-reduction approach is proposed for addressing nonlinear relationships within attributes. In this work, the attributes in the data are first organized into groups. In each group, the dimensions are reduced via a parametric mapping that takes into account nonlinear relationships. Mapping parameters are estimated using a low rank singular value decomposition (SVD) of distance covariance. Subsequently, the attributes are reorganized into groups based on the magnitude of their respective singular values. The group-wise organization and the subsequent reduction process is performed for multiple steps until a singular value-based user-defined criterion is satisfied. Simulation analysis is …

Direct Error Driven Learning For Deep Neural Networks With Applications To Bigdata, R. Krishnan, Jagannathan Sarangapani, V. A. Samaranayake

Electrical and Computer Engineering Faculty Research & Creative Works

In this paper, generalization error for traditional learning regimes-based classification is demonstrated to increase in the presence of bigdata challenges such as noise and heterogeneity. To reduce this error while mitigating vanishing gradients, a deep neural network (NN)-based framework with a direct error-driven learning scheme is proposed. To reduce the impact of heterogeneity, an overall cost comprised of the learning error and approximate generalization error is defined where two NNs are utilized to estimate the costs respectively. To mitigate the issue of vanishing gradients, a direct error-driven learning regime is proposed where the error is directly utilized for learning. It …

Preface Of 2nd Annual Meeting Of Siam Central States Section, Xiaoming He, Stephen Pankavich, Erik Van Vleck, Zhu Wang, Xiu Ye

Mathematics and Statistics Faculty Research & Creative Works

No abstract provided.

Approximation Degree Of Durrmeyer-Bézier Type Operators, Purshottam N. Agrawal, Serkan Araci, Martin Bohner, Kumari Lipi

Mathematics and Statistics Faculty Research & Creative Works

Recently, a mixed hybrid operator, generalizing the well-known Phillips operators and Baskakov-Szász type operators, was introduced. In this paper, we study Bézier variant of these new operators. We investigate the degree of approximation of these operators by means of the Lipschitz class function, the modulus of continuity, and a weighted space. We study a direct approximation theorem by means of the unified Ditzian-Totik modulus of smoothness. Furthermore, the rate of convergence for functions having derivatives of bounded variation is discussed.

Phytoforensics: Trees As Bioindicators Of Potential Indoor Exposure Via Vapor Intrusion, Jordan L. Wilson, V. A. Samaranayake, Matt A. Limmer, Joel Gerard Burken

Mathematics and Statistics Faculty Research & Creative Works

Human exposure to volatile organic compounds (VOCs) via vapor intrusion (VI) is an emerging public health concern with notable detrimental impacts on public health. Phytoforensics, plant sampling to semi-quantitatively delineate subsurface contamination, provides a potential non-invasive screening approach to detect VI potential, and plant sampling is effective and also time- and cost-efficient. Existing VI assessment methods are time- and resource-intensive, invasive, and require access into residential and commercial buildings to drill holes through basement slabs to install sampling ports or require substantial equipment to install groundwater or soil vapor sampling outside the home. Tree-core samples collected in 2 days at …

Mini Review: A Note On Nonoscillatory Solutions For Higher Dimensional Time Scale Systems, Elvan Akin, Ozkan Ozturk, Ismail Ugur Tiryaki, Gulsah Yeni

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we focus on nonoscillatory solutions of two (2D) and three (3D) dimensional time scale systems and discuss nonexistence of such solutions.

The Generalized Hypergeometric Difference Equation, Martin Bohner, Tom Cuchta

Mathematics and Statistics Faculty Research & Creative Works

A difference equation analogue of the generalized hypergeometric differential equation is defined, its contiguous relations are developed, and its relation to numerous well-known classical special functions are demonstrated.

A Higher-Order Ensemble/Proper Orthogonal Decomposition Method For The Nonstationary Navier-Stokes Equations, Max Gunzburger, Nan Jiang, Michael Schneier

Mathematics and Statistics Faculty Research & Creative Works

Partial differential equations (PDE) often involve parameters, such as viscosity or density. An analysis of the PDE may involve considering a large range of parameter values, as occurs in uncertainty quantification, control and optimization, inference, and several statistical techniques. The solution for even a single case may be quite expensive; whereas parallel computing may be applied, this reduces the total elapsed time but not the total computational effort. In the case of flows governed by the Navier-Stokes equations, a method has been devised for computing an ensemble of solutions. Recently, a reduced-order model derived from a proper orthogonal decomposition (POD) …

Parametrization Of Scale-Invariant Self-Adjoint Extensions Of Scale-Invariant Symmetric Operators, Miron B. Bekker, Martin Bohner, Alexander P. Ugol'nikov, Hristo Voulov

Mathematics and Statistics Faculty Research & Creative Works

On a Hilbert space H, we consider a symmetric scale-invariant operator with equal defect numbers. It is assumed that the operator has at least one scale invariant self-adjoint extension in H. We prove that there is a one-to-one correspondence between (generalized) resolvents of scale-invariant extensions and solutions of some functional equation. Two examples of Dirac-type operators are considered.

Decoupled, Linear, And Energy Stable Finite Element Method For The Cahn-Hilliard-Navier-Stokes-Darcy Phase Field Model, Yali Gao, Xiaoming He, Liquan Mei, Xiaofeng Yang

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we consider the numerical approximation for a phase field model of the coupled two-phase free flow and two-phase porous media flow. This model consists of Cahn—Hilliard—Navier—Stokes equations in the free flow region and Cahn—Hilliard—Darcy equations in the porous media region that are coupled by seven interface conditions. The coupled system is decoupled based on the interface conditions and the solution values on the interface from the previous time step. A fully discretized scheme with finite elements for the spatial discretization is developed to solve the decoupled system. In order to deal with the difficulties arising from the …

Incremental Proper Orthogonal Decomposition For Pde Simulation Data: Algorithms And Analysis, Hiba Fareed

Doctoral Dissertations

"We propose an incremental algorithm to compute the proper orthogonal decomposition (POD) of simulation data for a partial differential equation. Specifically, we modify an incremental matrix SVD algorithm of Brand to accommodate data arising from Galerkin-type simulation methods for time dependent PDEs. We introduce an incremental SVD algorithm with respect to a weighted inner product to compute the proper orthogonal decomposition (POD). The algorithm is applicable to data generated by many numerical methods for PDEs, including finite element and discontinuous Galerkin methods. We also modify the algorithm to initialize and incrementally update both the SVDand an error bound during the …

Cox-Type Model Validation With Recurrent Event Data, Muna Mohamed Hammuda

Doctoral Dissertations

"Recurrent event data occurs in many disciplines such as actuarial science, biomedical studies, sociology, and environment to name a few. It is therefore important to develop models that describe the dynamic evolution of the event occurrences. One major problem of interest to researchers with these types of data is models for the distribution function of the time between events occurrences, especially in the presence of covariates that play a major role in having a better understanding of time to events.

This work pertains to statistical inference of the regression parameter and the baseline hazard function in a Cox-type model for …

Hdg Methods For Dirichlet Boundary Control Of Pdes, Yangwen Zhang

Doctoral Dissertations

"We begin an investigation of hybridizable discontinuous Galerkin (HDG) methods for approximating the solution of Dirichlet boundary control problems for PDEs. These problems can involve atypical variational formulations, and often have solutions with low regularity on polyhedral domains. These issues can provide challenges for numerical methods and the associated numerical analysis. In this thesis, we use an existing HDG method for a Dirichlet boundary control problem for the Poisson equation, and obtain optimal a priori error estimates for the control in the high regularity case. We also propose a new HDG method to approximate the solution of a Dirichlet boundary …

On Modeling Quantities For Insurer Solvency Against Catastrophe Under Some Markovian Assumptions, Daniel Jefferson Geiger

Doctoral Dissertations

"Insurance companies sometimes face catastrophic losses, yet they must remain solvent enough to meet the legal obligation of covering all claims. Catastrophes can result in large damages to the policyholders, causing the arrival of numerous claims to insurance companies at once. Furthermore, the severity of an event could impact the time until the next occurrence. An insurer needs certain levels of startup capital to meet all claims, and then must have adequate reserves on a continual basis, even more so when catastrophes occur. This work examines two facets of these matters: for an infinite time horizon, we extend and develop …

Numerical Modeling Of Capillary-Driven Flow In Open Microchannels: An Implication Of Optimized Wicking Fabric Design, Mehrad Gholizadeh Ansari

Masters Theses

"The use of microfluidics to transfer fluids without applying any exterior energy source is a promising technology in different fields of science and engineering due to their compactness, simplicity and cost-effective design. In geotechnical engineering, to increase the soil's strength, hydrophilic wicking fibers as type of microfluidics have been employed to transport and drain water out of soil spontaneously by taking advantage of natural capillary force without using any pumps or other auxiliary devices. The objective of this study is to understand the scientific mechanisms of the capability for wicking fiber to drain both gravity and capillary water out of …

Convergence Analysis And Error Estimates For A Second Order Accurate Finite Element Method For The Cahn–Hilliard–Navier–Stokes System, Amanda E. Diegel, Cheng Wang, Xiaoming Wang, Steven M. Wise

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we present a novel second order in time mixed finite element scheme for the Cahn–Hilliard–Navier–Stokes equations with matched densities. the scheme combines a standard second order Crank–Nicolson method for the Navier–Stokes equations and a modification to the Crank–Nicolson method for the Cahn–Hilliard equation. in particular, a second order Adams-Bashforth extrapolation and a trapezoidal rule are included to help preserve the energy stability natural to the Cahn–Hilliard equation. We show that our scheme is unconditionally energy stable with respect to a modification of the continuous free energy of the PDE system. Specifically, the discrete phase variable is shown …

Magnetic Control Of Lateral Migration Of Ellipsoidal Microparticles In Microscale Flows, R. Zhou, C. A. Sobecki, J. Zhang, Yanzhi Zhang, Cheng Wang

Mathematics and Statistics Faculty Research & Creative Works

No abstract provided.

Hypoelliptic Multiscale Langevin Diffusions: Large Deviations, Invariant Measures And Small Mass Asymptotics, Wenqing Hu, Konstantinos Spiliopoulosï

Mathematics and Statistics Faculty Research & Creative Works

We consider a general class of hypoelliptic Langevin diffusions and study two related questions. The first question is large deviations for hypoelliptic multiscale diffusions as the noise and the scale separation parameter go to zero. The second question is small mass asymptotics of (a) the invariant measure corresponding to the hypoelliptic Langevin operator and of (b) related hypoelliptic Poisson equations. The invariant measure corresponding to the hypoelliptic problem and appropriate hypoelliptic Poisson equations enter the large deviations rate function due to the multiscale effects. Based on the small mass asymptotics we derive that the large deviations behavior of the multiscale …

Oscillation Criteria For Third-Order Nonlinear Functional Difference Equations With Damping, Martin Bohner, C. Dharuman, R. Srinivasan, Ethiraju Thandapani

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we obtain some new criteria for the oscillation of certain third-order difference equations using comparison principles with a suitable couple of first-order difference equations. The presented results improve and extend the earlier ones. Examples are provided to illustrate the main results.

Stationary Acceleration Of Frenet Curves, Nemat Abazari, Martin Bohner, Ilgin Sager, Yusuf Yayli

Mathematics and Statistics Faculty Research & Creative Works

In this paper, the stationary acceleration of the spherical general helix in a 3-dimensional Lie group is studied by using a bi-invariant metric. The relationship between the Frenet elements of the stationary acceleration curve in 4-dimensional Euclidean space and the intrinsic Frenet elements of the Lie group is outlined. As a consequence, the corresponding curvature and torsion of these curves are computed. In Minkowski space, for the curves on a timelike surface to have a stationary acceleration, a necessary and sufficient condition is refined.

Itô'S Formula, The Stochastic Exponential, And Change Of Measure On General Time Scales, Wenqing Hu

Mathematics and Statistics Faculty Research & Creative Works

We provide an Itô formula for stochastic dynamical equation on general time scales. Based on this Itô's formula we give a closed-form expression for stochastic exponential on general time scales. We then demonstrate Girsanov's change of measure formula in the case of general time scales. Our result is being applied to a Brownian motion on the quantum time scale (𝑞-time scale).

Founders, Feminists, And A Fascist -- Some Notable Women In The Missouri Section Of The Maa, Leon M. Hall

Mathematics and Statistics Faculty Research & Creative Works

In the history of the Missouri Section of the MAA, some of the more interesting people who influenced the growth and development of the section through the years were and are women. In this chapter, we discuss the contributions of a few (certainly not all) of these women to the Missouri Section and mathematics as a whole, including Emily Kathryn Wyant (founder of KME), Margaret F. Willerding (who dealt with sexism in the 1940s), Maria Castellani (an official in Mussolini’s Italy before coming to America), and T. Christine Stevens (co-founder of Project NExT). Without them, and others like them, both …

Chain Transitive Homeomorphisms On A Space: All Or None, Ethan Akin, Juho Rautio

Mathematics and Statistics Faculty Research & Creative Works

Extending earlier work, we consider when a compact metric space can be realized as the omega limit set of a discrete time dynamical system. This is equivalent to asking when the space admits a chain transitive homeomorphism. We approach this problem in terms of various conditions on the connected components of the space. We also construct spaces where all homeomorphisms are chain transitive.

A Harmonic M-Factorial Function And Applications, Reginald Alfred Brigham Ii

Doctoral Dissertations

"We offer analogs to the falling factorial and rising factorial functions for the set of harmonic numbers, as well as a mixed factorial function called the M-factorial. From these concepts, we develop a harmonic analog of the binomial coefficient and an alternate expression of the harmonic exponential function and establish several identities. We generalize from the harmonic numbers to a general time scale and demonstrate how solutions to some second order eigenvalue problems and partial dynamic equations can be constructed using power series built from the M-factorial function"--Abstract, page iii.

Bootstrap-Based Confidence Intervals In Partially Accelerated Life Testing, Ahmed Mohamed Eshebli

Doctoral Dissertations

"Accelerated life testing (ALT) is utilized to estimate the underlying failure distribution and related parameters of interest in situations where the components under study are designed for long life and therefore will not yield failure data within a reasonable test period. In ALT, life testing is carried out under two or more higher than normal stress levels, with the resulting acceleration of the failure process yielding a sufficient amount of un-censored life-span data within a practical test duration. Usually one (or more) parameters of the life distribution is linked to the stress level through a suitably selected model based on …

Programming Problems On Time Scales: Theory And Computation, Rasheed Basheer Al-Salih

Doctoral Dissertations

"In this dissertation, novel formulations for several classes of programming problems are derived and proved using the time scales technique. The new formulations unify the discrete and continuous programming models and extend them to other cases "in between." Moreover, the new formulations yield the exact optimal solution for the programming problems on arbitrary isolated time scales, which solve an important open problem. Throughout this dissertation, six distinct classes of programming problems are presented as follows. First, the primal as well as the dual time scales linear programming models on arbitrary time scales are formulated. Second, separated linear programming primal and …

Balanced Truncation Model Reduction Of Nonlinear Cable-Mass Pde System, Madhuka Hareena Lochana Weerasinghe

Doctoral Dissertations

We consider model order reduction of a cable-mass system modeled by a one dimensional wave equation with interior damping and dynamic boundary conditions. The system is driven by a time dependent forcing input to a linear mass-spring system at the left boundary of the cable. A mass-spring model at the right end of the cable includes a nonlinear stiffening force. The goal of the model reduction is to produce a low order model that produces an accurate approximation to the displacement and velocity of the mass in the nonlinear mass-spring system at the right boundary. We believe the nonlinear cable-mass …