Mathematics Commons^™

Mathematics For Biomedical Physics, Jogindra M. Wadehra

Open Textbooks

Mathematics for Biomedical Physics is an open access peer-reviewed textbook geared to introduce several mathematical topics at the rudimentary level so that students can appreciate the applications of mathematics to the interdisciplinary field of biomedical physics. Most of the topics are presented in their simplest but rigorous form so that students can easily understand the advanced form of these topics when the need arises. Several end-of-chapter problems and chapter examples relate the applications of mathematics to biomedical physics. After mastering the topics of this book, students would be ready to embark on quantitative thinking in various topics of biology and …

The Local Cohomology Spectral Sequence For Topological Modular Forms, Robert Bruner, John Greenlees, John Rognes

Mathematics Faculty Research Publications

We discuss proofs of local cohomology theorems for topological modular forms, based on Mahowald–Rezk duality and on Gorenstein duality, and then make the associated local cohomology spectral sequences explicit, including their differential patterns and hidden extensions.

The Adams Spectral Sequence For The Image-Of-J Spectrum, Robert R. Bruner, John Rognes

Mathematics Faculty Research Publications

We show that if we factor the long exact sequence in cohomology of a cofiber sequence of spectra into short exact sequences, then the d_2-differential in the Adams spectral sequence of any one term is related in a precise way to Yoneda composition with the 2-extension given by the complementary terms in the long exact sequence. We use this to give a complete analysis of the Adams spectral sequence for the connective image-of-J spectrum, finishing a calculation that was begun by D. Davis [Bol. Soc. Mat. Mexicana (2) 20 (1975), pp. 6–11].

The Cohomology Of The Mod 2 Steenrod Algebra, Robert R. Bruner, John Rognes

Open Data at Wayne State

The dataset contains a minimal resolution of the mod 2 Steenrod algebra in the range 0 <= s <= 128, 0 <= t <= 200, together with chain maps for each cocycle in that range and for the squaring operation Sq^0 in the cohomology of the Steenrod algebra. The included document CohomA2.pdf explains the contents and usage of the dataset in detail (also available as supplemental material in this record).

Dataset is also available at the NIRD Research Data Archive, https://doi.org/10.11582/2021.00077; Data Description also available at arXiv.org, https://doi.org/10.48550/arXiv.2109.13117.

Integral Representations Of Sl_2(Z/Nz), Yatin Dinesh Patel

Wayne State University Dissertations

The aim of this work is to determine for which commutative rings integral representations of SL_2(Z/nZ) exist and to explicitly compute them. We start with R = Z/pZ and then consider Z=p^\lambda Z. A new approach will be used to do this based on the Weil representation. We then consider general finite rings Z/nZ by extending methods described in [26]. We make extensive use of group theory, linear representations of finite groups, ring theory, algebraic geometry, and number theory. From number theory we will employ results regarding modular forms, Legendre symbols, Hilbert symbols, and quadratic forms. We consider the works …

ℂ-Motivic Modular Forms, Bogdan Gheorghe, Daniel C. Isaksen, Achim Krause, Nicolas Ricka

Mathematics Faculty Research Publications

We construct a topological model for cellular, 2-complete, stable C-motivic homotopy theory that uses no algebro-geometric foundations.We compute the Steenrod algebra in this context, and we construct a “motivic modular forms” spectrum over ℂ.

From Mathematics To Medicine: A Practical Primer On Topological Data Analysis (Tda) And The Development Of Related Analytic Tools For The Functional Discovery Of Latent Structure In Fmri Data, Andrew Salch, Adam Regalski, Hassan Abdallah, Raviteja Suryadevara, Michael J. Catanzaro, Vaibhav A. Diwadkar

Mathematics Faculty Research Publications

fMRI is the preeminent method for collecting signals from the human brain in vivo, for using these signals in the service of functional discovery, and relating these discoveries to anatomical structure. Numerous computational and mathematical techniques have been deployed to extract information from the fMRI signal. Yet, the application of Topological Data Analyses (TDA) remain limited to certain sub-areas such as connectomics (that is, with summarized versions of fMRI data). While connectomics is a natural and important area of application of TDA, applications of TDA in the service of extracting structure from the (non-summarized) fMRI data itself are heretofore nonexistent. …

The Adams Spectral Sequence For Topological Modular Forms, Robert Bruner, John Rognes

Mathematics Faculty Research Publications

The connective topological modular forms spectrum, 𝑡𝑚𝑓, is in a sense initial among elliptic spectra, and as such is an important link between the homotopy groups of spheres and modular forms. A primary goal of this volume is to give a complete account, with full proofs, of the homotopy of 𝑡𝑚𝑓 and several 𝑡𝑚𝑓-module spectra by means of the classical Adams spectral sequence, thus verifying, correcting, and extending existing approaches. In the process, folklore results are made precise and generalized. Anderson and Brown-Comenetz duality, and the corresponding dualities in homotopy groups, are carefully proved. The volume also includes an account …

The Hatcher-Quinn Invariant And Differential Forms, Joshua Lenwood Turner

Wayne State University Dissertations

An intersection problem consists of submanifolds $P, Q \subset M$ having non-empty intersection. In 1974, Hatcher and Quinn introduced a bordism-theoretic obstruction to finding a deformation of $P$ off of $Q$ by an isotopy. This dissertation studies the problem of finding an analytical expression for the Hatcher-Quinn obstruction---one which involves the language of differential forms. We first introduce the notion of a smooth structure on a set by introducing a system of mappings called plots. By generalizing this to the fibered setting, we use the concept to give a model for the homology of the generalized path space $E$ i.e., …

Discrete-Time Control With Non-Constant Discount Factor, Héctor Jasso-Fuentes, José-Luis Menaldi, Tomás Prieto-Rumeau

Mathematics Faculty Research Publications

This paper deals with discrete-time Markov decision processes (MDPs) with Borel state and action spaces, and total expected discounted cost optimality criterion. We assume that the discount factor is not constant: it may depend on the state and action; moreover, it can even take the extreme values zero or one. We propose sufficient conditions on the data of the model ensuring the existence of optimal control policies and allowing the characterization of the optimal value function as a solution to the dynamic programming equation. As a particular case of these MDPs with varying discount factor, we study MDPs with stopping, …

The Wedge Family Of The Cohomology Of The C-Motivic Steenrod Algebra, Hieu Trung Thai

Wayne State University Dissertations

Computing the stable homotopy groups of the sphere spectrum is one of the most important problems of stable homotopy theory. Focusing on the 2-complete stable homotopy groups instead of the integral homotopy groups, the Adams spectral sequence appears to be one of the most effective tools to compute the homotopy groups. The spectral sequence has been studied by J. F. Adams, M. Mahowald, M. Tangora, J. P. May and others.

In 1999, Morel and Voevodsky introduced motivic homotopy theory. One of its consequences is the realization that almost any object studied in classical algebraic topology could be given a motivic …

Variational Analysis In Second-Order Cone Programming And Applications, Hang Thi Van Nguyen

Wayne State University Dissertations

This dissertation conducts a second-order variational analysis for an important class on nonpolyhedral conic programs generated by the so-called second-order/Lorentz/ice-cream cone. These second-order cone programs (SOCPs) are mathematically challenging due to the nonpolyhedrality of the underlying second-order cone while being important for various applications. The two main devices in our study are second epi-derivative and graphical derivative of the normal cone mapping which are proved to accumulate vital second-order information of functions/constraint systems under investigation. Our main contribution is threefold:

- proving the twice epi-differentiability of the indicator function of the second-order cone and of the augmented Lagrangian associated with …

Study Of Grain Growth In Single-Phase Polycrystals, Pawan Vedanti

Wayne State University Dissertations

Materials with random microstructure are characterized by additional thermodynamic parameters, entropy and temperature of microstructure. It has been argued that there is one more law of thermodynamics: entropy of microstructure decays in isolated systems. This assertion has been checked experimentally for the process of grain growth which showed that entropy of grain structure decays indeed as expected. The equation of state for microstructure entropy has also been studied. In general, entropy of grain microstructure is expected to be a function of grain structure energy and the average grain size. Our experiments suggest that in fact, the equation of state degenerates …

Making Real-World Connections In High School Mathematics: The Effectiveness Of A Professional Development Program In Changing Teachers’ Knowledge, Beliefs, And Practices, Thad Ludlam Wilhelm

Wayne State University Dissertations

The study aimed to assess the impact of a professional development workshop at changing secondary mathematics teachers’ knowledge, beliefs, and practices related to real-world applications of algebra. It also addressed gaps in the research literature related to teacher knowledge of how algebra is used by professionals in non-academic settings and their beliefs about the relevance of algebra to their students’ lives. The observational study employed mixed methods. Principal components analysis was conducted on responses to an online questionnaire. Pre-test vs. post-test comparisons were made for workshop participants. Treatment vs. control comparisons were also made using a nationally representative random sample …

Stochastic Approximation And Applications To Networked Systems, Thu Thi Le Nguyen

Wayne State University Dissertations

This dissertation focuses on a class of SA algorithms with applications to networked systems and is based on the published works that have been done jointly during my Ph.D. training. The networked systems are fundamentally characterized by interaction among control, communications, and computing, with applications in a vast array of emerging technologies such as smart grids, intelligent transportation systems, social networks, smart city, to name just a few. Networked systems encounter many environment uncertainties that are inherently stochastic. Besides the aforementioned advantages, the framework of SA can also accommodate multiple random processes and diversified system dynamics, even random and distributed …

On Optimal Stopping And Impulse Control With Constraint, J. L. Menaldi, M. Robin

Mathematics Faculty Research Publications

The optimal stopping and impulse control problems for a Markov-Feller process are considered when the controls are allowed only when a signal arrives. This is referred to as control problems with constraint. In [28, 29, 30], the HJB equation was solved and an optimal control (for the optimal stopping problem, the discounted impulse control problem and the ergodic impulse control problem, respectively) was obtained, under suitable conditions, including a setting on a compact metric state space. In this work, we extend most of the results to the situation where the state space of the Markov process is locally compact.

Well-Posedness And Symmetry Properties Of Free Boundary Problems For Some Non-Linear Degenerate Elliptic Second Order Partial Differential Equations, Alaa Haj Ali

Wayne State University Dissertations

In the first part of this thesis, a bifurcation about the uniqueness of a solution of a singularly perturbed free boundary problem of phase transition associated with the $p$-Laplacian, subject to given boundary condition is proved in the first chapter. We show this phenomenon by proving the existence of a third solution through the Mountain Pass Lemma when the boundary data decreases below a threshold. In the second chapter and third chapter, we prove the convergence of an evolution to stable solutions, and show the Mountain Pass solution is unstable in this sense.

In the second part of this thesis, …

Teachers' Reflection On Their Beliefs And Question-Asking Practices During Mathematics Instruction, Kaili Takiyah Hardamon

Wayne State University Dissertations

Teachers’ daily instructional practices are a critical component in creating a rich and meaningful educational experience for students. Thus, factors that inform instructional practices are of particular importance and interest to education researchers and other stakeholders. Beliefs about teaching and learning are a known factor influencing teachers’ instructional practices (Ernest, 1989). This study focused on a specific instructional practice, question-asking, which has a profound impact on students’ experience with mathematics (Weiland, Hudson, and Amador (2014). Understanding the relationship between teachers’ beliefs and practice helps to make sense of teachers’ decision-making processes, particularly as they choose questions to ask students during …

Second-Order Generalized Differentiation Of Piecewise Linear-Quadratic Functions And Its Applications, Hong Do

Wayne State University Dissertations

The area of second-order variational analysis has been rapidly developing during the recent years with many important applications in optimization. This dissertation is devoted to the study and applications of the second-order generalized differentiation of a remarkable

class of convex extended-real-valued functions that is highly important in many aspects of nonlinear and variational analysis, specifically those related to optimization and stability.

The first goal of this dissertation is to compute the second-order subdifferential of the functions described above, which will be applied in the study of the stability of composite optimization problems associated with piecewise linear-quadratic functions, known as extended …

Numerical Approaches To A Thermoelastic Kirchhoff-Love Plate System, Zeyu Zhou

Wayne State University Dissertations

In this work, theory background of the sobolev spaces and finite element spaces are

reviewed first. Then the details of how the thermoelastic Kirchhoff-Love(KL) plates numerically established are presented. Later we approaches to the thermoelastic KL system numerically with mixed element method, H^1−Galerkin method and interior penalty discontinuous galerkin method(IP-DG).

What is more, the SIP-DG also applied to solve this KL system numerically. The well-posedness, existence, uniqueness and convergence properties are theoretical analyzed. The gain of the convergence rate is also O(h^k), that is 1 less than the observed convergence rate.

When discussing the H1-Galerkin method, the main advantages over …

Spectral Methods For Hamiltonian Systems And Their Applications, Lewei Zhao

Wayne State University Dissertations

Hamiltonian systems typically arise as models of conservative physical systems and have many applications. Our main emphasis is using spectral methods to preserve both symplectic structure and energy up to machine error in long time. An engery error estimation is given for a type of Hamiltonian systems with polynomial nonlinear part, which is numerical verified by solving a Henon-Heiles systems. Three interesting applications are presented : the first one is the N-body problems. The second one is approximation for Weyl's Law and the third one is simulating quantum cooling in an optomechanical system to study the dissipative dynamics. Moreover, nonsmooth …

Switching Diffusion Systems With Past-Dependent Switching Having A Countable State Space, Hai Dang Nguyen

Wayne State University Dissertations

Emerging and existing applications in wireless communications, queueing networks, biological models, financial engineering, and social networks demand the

mathematical modeling and analysis of hybrid models in which continuous dynamics and discrete events coexist.

Assuming that the systems are in continuous times,

stemming from stochastic-differential-equation-based models and random discrete events,

switching diffusions come into being. In such systems, continuous states and discrete events

(discrete states)

coexist and interact.

A switching diffusion is a two-component process $(X(t),\alpha(t))$, a continuous component and a discrete component taking values in a discrete set (a set consisting of isolated points).

When the discrete component takes a …

Hardy Space Theory And Endpoint Estimates For Multi-Parameter Singular Radon Transforms, Jiawei Shen

Wayne State University Dissertations

In [12], Christ, Nagel, Stein and Waigner studied the L p theories for the singular Radon Trans-

forms. Furthermore, B. Street in [68], and Stein and Street in [64–67] extended the theories of the

L p boundedness for multi-parameter singular integral operators, such as the Calderón Zygmund

operators and singular Radon transforms. In this dissertation, we will study the Hardy space H p

and its dual space associated with both the one-parameter and multi-parameter singular Radon

transforms, and consider the boundedness of the singular Radon transforms on such Hardy spaces

H p when 0 ≤ p ≤ 1.

Inspired by …

Hybrid Stochastic Systems: Numerical Methods, Limit Results, And Controls, Tuan A. Hoang

Wayne State University Dissertations

This dissertation is concerned with the so-called stochastic hybrid systems, which are

featured by the coexistence of continuous dynamics and discrete events and their interactions. Such systems have drawn much needed attentions in recent years. One of the main reasons is that such systems can be used to better reflect the reality for a wide range of applications in networked systems, communication systems, economic systems, cyber-physical systems, and biological and ecological systems, among others. Our main interest is centered around one class of such hybrid systems known as switching diffusions. In such a system, in addition to the driving force …

The Motivic Cofiber Of Τ And Exotic Periodicities, Bogdan Gheorghe

Wayne State University Dissertations

Consider the Tate twist τ ∈ H 0,1 (S 0,0 ) in the mod 2 cohomology of the motivic sphere.

After 2-completion, the motivic Adams spectral sequence realizes this element as a map

τ : S 0,−1 GGA S 0,0 . This thesis begins with the study of its cofiber, that we denote by Cτ.

We first show that this motivic 2-cell complex can be endowed with a unique E ∞ ring

structure. This promotes the known isomorphism π ∗,∗ Cτ ∼= Ext ∗,∗ BP ∗ BP (BP ∗ ,BP ∗ )

to an isomorphism of rings which also preserves …

Polynomial Preserving Recovery For Weak Galerkin Methods And Their Applications, Ren Zhao

Wayne State University Dissertations

Gradient recovery technique is widely used to reconstruct a better numerical gradient from a finite element solution, for mesh smoothing,

a posteriori error estimate and adaptive finite element methods. The PPR technique generates a higher order approximation of the gradient on a patch of mesh elements around each mesh vertex. It can be used for different finite element methods for different problems. This dissertation presents recovery techniques for the weak Galerkin methods and as well as applications of gradient recovery on various of problems, including elliptic problems, interface problems, and Stokes problems.

Our first target is to develop a boundary …

Periodicity In Iterated Algebraic K-Theory Of Finite Fields, Gabriel Angelini-Knoll

Wayne State University Dissertations

In this dissertation, we study the interactions between periodic phenomena in the homotopy groups of spheres and algebraic K-theory of ring spectra. C. Ausoni and J. Rognes initiated a program to study the arithmetic of ring spectra using algebraic K-theory and gave a higher chromatic version of the Lichtenbaum-Quillen conjecture, called the red-shift conjecture, that is expected to govern this arithmetic. This dissertation provides a proof of a special case of a variation on the red-shift conjecture. Specifically, we show that, under conditions on the order of the fields, iterated algebraic K-theory of finite fields detects a periodic family chromatic …

Measure And Integration, Jose L. Menaldi

Mathematics Faculty Research Publications

Abstract measure and integration, with theory and (solved) exercises is developed. Parts of this book can be used in a graduate course on real analysis.

Distributions And Function Spaces, Jose L. Menaldi

Mathematics Faculty Research Publications

Beginning with a quick recall on measure and integration theory, basic concepts on (a) Function Spaces, (b) Schwartz Theory of Distributions, and (c) Sobolev and Besov Spaces are developed. Moreover, only a few number of (solved) exercises are given. Parts of this book can be used in a graduate course on real analysis.

A Counterexample For Lightning Flash Modules Over E(E1,E2), David Benson, Robert R. Bruner

Mathematics Faculty Research Publications

We give a counterexample to Theorem 5 in Section 18.2 of Margolis’ book, “Spectra and the Steenrod Algebra” and make remarks about the proofs of some later theorems in the book that depend on it. The counterexample is a module which does not split as a sum of lightning flash modules and free modules.