Mathematics Commons^™

Community Detection In Complex Networks, Zhenqi Lu

McKelvey School of Engineering Graduate Student Theses & Dissertations

Network science plays a central role in understanding and modeling complex systems in many disciplines, including physics, sociology, biology, computer science, economics, politics, and neuroscience. By studying networks, we can gain a deep understanding of the behavior of the systems they represent. Many networks exhibit community structure, i.e., they have clusters of nodes that are locally densely interconnected. These communities manifest the hierarchical organization of the objects in systems, and detecting communities greatly facilitates the study of the organization and structure of complex systems.

Most existing community-detection methods consider low-order connection patterns, at the level of individual links. But high-order …

Smooth Ica Under Time Pattern Assumptions, Jiayi Fu

Arts & Sciences Graduate Student Theses and Dissertations

Independent component analysis (ICA) is wildly used in differently areas. As traditional ICA models make no assumptions on time pattern, they do not take time domain information into consideration. In this thesis, we introduced new assumptions that allow local dependence over time, and we built smooth ICA models to utilize the smoothness information for sources signals. Based on the local dependence assumptions, constrained optimization problems with smoothing penalty were discussed. Then we introduced smooth ICA estimators and estimating equations. Under local dependence assumptions, we gave proofs about the consistency and asymptotic normality of these estimators. We derived the Newton iterative …

Inhibition Of Aminoglycoside 6’-N-Acetyltransferase Type Ib [Aac(6′)-Ib]: Structure-Activity Relationship Of Substituted Pyrrolidine Pentamine Derivatives As Inhibitors, Kenneth Rocha, Jesus Magallon, Craig Reeves, Kimberly Phan, Peter Vu, Crista L. Oakley-Havens, Stella Kwan, Maria S. Ramirez, Travis Lavoi, Haley Donow, Prem Chapagain, Radleigh Santos, Clemencia Pinilla, Marc A. Giulianotti, Marcelo E. Tolmasky

Mathematics Faculty Articles

The aminoglycoside 6′-N-acetyltransferase type Ib [AAC(6′)-Ib] is a common cause of resistance to amikacin and other aminoglycosides in Gram-negatives. Utilization of mixture-based combinatorial libraries and application of the positional scanning strategy identified an inhibitor of AAC(6′)-Ib. This inhibitor’s chemical structure consists of a pyrrolidine pentamine scaffold substituted at four locations (R1, R3, R4, and R5). The substituents are two S-phenyl (R1 and R4), an S-hydroxymethyl (R3), and a 3-phenylbutyl (R5) groups. Another location, R2, does not have a substitution, but it is named because its stereochemistry was modified in some compounds utilized in this study. Structure-activity …

The Life And Scientific Work Of Hiroshi Kunita, Yasushi Ishikawa

Journal of Stochastic Analysis

No abstract provided.

Lag Time Between State-Level Policy Interventions And Change Points In Covid-19 Outcomes In The United States, Jaechoul Lee

Mathematics Faculty Publications and Presentations

State-level policy interventions have been critical in managing the spread of the new coronavirus. Here, we study the lag time between policy interventions and change in COVID-19 outcome trajectory in the United States. We develop a stepwise drifts random walk model to account for non-stationarity and strong temporal correlation and subsequently apply a change-point detection algorithm to estimate the number and times of change points in the COVID-19 outcome data. Furthermore, we harmonize data on the estimated change points with non-pharmaceutical interventions adopted by each state of the United States, which provides us insights regarding the lag time between the …

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. …

Topology Optimization Of 2d Structures With Multiple Displacement Constraints, Patricio Uarac Pinto

International Programs

The use of topology optimization in the design process in Civil Engineering can lead to innovative building shapes that not only fulfill structural requirements but also open new opportunities for arquitectonics.

Lower Bounds On Betti Numbers, Adam Boocher, Eloisa Grifo

Department of Mathematics: Faculty Publications

We survey recent results on bounds for Betti numbers of modules over polynomial rings, with an emphasis on lower bounds. Along the way, we give a gentle introduction to free resolutions and Betti numbers, and discuss some of the reasons why one would study these.

Mapping Polynomial Dynamics, Devin Becker

DePaul Discoveries

We explore the complex dynamics of a family of polynomials defined on the complex plane by f(z) = az^m(1+z/d)^d where a is a complex number not equal to zero, and m and d are at least 2. These functions have three finite critical points, one of which has behavior that differs as we change our parameter values. We analyze the dynamical behavior at this critical point, with a particular interest in the structures that appear in the filled Julia set K(f) and the basin of infinity A_{\infty}(f). The behavior of the family is extremely sensitive to our …

Triangulation And Finite Element Method For A Variational Problem Inspired By Medical Imaging, Tim Komperda, Enrico Au-Yeung

DePaul Discoveries

We implement the finite element method to solve a variational problem that is inspired by medical imaging. In our application, the domain of the image does not need to be a rectangle and can contain a cavity in the middle. The standard approach to solve a variational problem involves formulating the problem as a partial differential equation. Instead, we solve the variational problem directly, using only techniques available to anyone familiar with vector calculus. As part of the computation, we also explore how triangulation is a useful tool in the process.

Sparse Domination Of The Martingale Transform, Michael Scott Kutzler

Mathematics & Statistics ETDs

Linear operators are of huge importance in modern harmonic analysis. Many operators can be dominated by finitely many sparse operators. The main result in this thesis is showing a toy operator, namely the Martingale Transform is dominated by a single sparse operator. Sparse operators are based on a sparse family which is simply a subset of a dyadic grid. We also show the A2 conjecture for the Martingale Transform which follows from the sparse domination of the Martingale Transform and the A2 conjecture for sparse operators.

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Memories Of Professor Hiroshi Kunita, Ichiro Shigkeawa

Journal of Stochastic Analysis

No abstract provided.

On The Works Of Hiroshi Kunita In The Sixties, Masatoshi Fukushima

Journal of Stochastic Analysis

No abstract provided.

Personal Memories Of Hiroshi Kunita, David Elworthy

Journal of Stochastic Analysis

No abstract provided.

Preface, Shigeki Aida, David Applebaum, Yasushi Ishikawa, Arturo Kohatsu-Higa, Nicolas Privault

Journal of Stochastic Analysis

No abstract provided.

An Introduction To Calling Bullshit: Learning To Think Outside The Black Box, Jevin D. West, Carl T. Bergstrom

Numeracy

Bergstrom, Carl T. and Jevin D. West. 2020. Calling Bullshit: The Art of Skepticism in a Data-Driven World. (New York: Random House) 336 pp. ISBN 978-0525509202.

While statistical methods receive greater attention, the art of critically evaluating information in everyday life more commonly depends on thinking outside the black box of the algorithm. In this piece we introduce readers to our book and associated online teaching materials—for readers who want to more capably call “bullshit” or to teach their students to do the same.

For The Love Of Mathematical Research: A Conversation With Undergraduate Research Students, Mike Janssen

Faculty Work Comprehensive List

"One of my passions as a professor is creating opportunities for students to ask questions about mathematics."

Posting about students' perspectives on mathematics research from In All Things - an online journal for critical reflection on faith, culture, art, and every ordinary-yet-graced square inch of God’s creation.

https://inallthings.org/for-the-love-of-mathematical-research-a-conversation-with-undergraduate-research-students/

Physics-Informed Structure-Preserving Numerical Approximations Of Thermodynamically Consistent Models For Non-Equilibrium Phenomena, Jia Zhao

Funded Research Records

No abstract provided.

Beurling-Lax Type Theorems And Cuntz Relations, Daniel Alpay, Fabrizio Colombo, Irene Sabadini, Baruch Schneider

Mathematics, Physics, and Computer Science Faculty Articles and Research

We prove various Beurling-Lax type theorems, when the classical backward-shift operator is replaced by a general resolvent operator associated with a rational function. We also study connections to the Cuntz relations. An important tool is a new representation result for analytic functions, in terms of composition and multiplication operators associated with a given rational function. Applications to the theory of de Branges-Rovnyak spaces, also in the indefinite metric setting, are given.

Ca Special Issue On Neural Network Approximation, Wolfgang Dahmen, Ronald A. Devore, Philipp Grohs

Faculty Publications

No abstract provided.

On Graphs With Proper Connection Number 2, Jill Faudree, Leah Berman, Glenn Chappell, Chris Hartman, John Gimbel, Gordon Williams

Theory & Applications of Graphs

An edge-colored graph is properly connected if for every pair of vertices u and v there exists a properly colored uv-path (i.e. a uv-path in which no two consecutive edges have the same color). The proper connection number of a connected graph G, denoted pc(G), is the smallest number of colors needed to color the edges of G such that the resulting colored graph is properly connected. An edge-colored graph is flexibly connected if for every pair of vertices u and v there exist two properly colored paths between them, say P and Q, such …

Mathematical Models And Homogenization Of Deer Dispersal, Environmental Hazard, And Direct/Indirect Transmission To Predict Spread Of Chronic Wasting Disease, James A. Powell

Funded Research Records

No abstract provided.

Flags And Tech To Teach Math, Kelly W. Remijan

Publications & Research

Teachers will discover innovative activities for math concepts that engage students with flags & technology such as CNC Machines, Google Sheets Pixel Art, Desmos, & GeoGebra.

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 …

Predicting (Economic) Trends: Why Signature Method In Machine Learning, Vladik Kreinovich, Chon Van Le

Departmental Technical Reports (CS)

In many practical situations, we can predict the trend -- i.e., how the system will change -- but we cannot predict the exact timing of this change: this timing may depend on many unpredictable factors. For example, we may be sure that the economy will recover, but how fast it will recover may depend on the status of the pandemic, on the weather-affected agriculture input, etc. In such trend predictions, one of the most efficient methods is signature method, which is based on applying machine learning techniques to several special characteristics of the corresponding time series. In this paper, we …

How To Work? How To Study? Shall We Cram For The Exams? And How Is This Related To Life On Earth?, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

If we follow the same activity for a long time, our productivity decreases. To increase productivity, a natural idea is therefore to switch to a different activity, and then to switch back and resume the current task. On the other hand, after each switch, we need some time to get back to the original productivity. As a result, too frequent switches are also counterproductive. Natural questions are: shall we switch? if yes, when? In this paper, we use a simple model to provide approximate answers to these questions.

Correcting Interval-Valued Expert Estimates: Empirical Formulas Explained, Laura A. Berrout Ramos, Vladik Kreinovich, Kittawit Autchariyapanitkul

Departmental Technical Reports (CS)

Experts' estimates are approximate. To make decisions based on these estimates, we need to know how accurate these estimate are. Sometimes, experts themselves estimate the accuracy of their estimates -- by providing the interval of possible values instead of a single number. In other cases, we can gauge the accuracy of the experts' estimates by asking several experts to estimates the same quantity and using the interval range of these values. In both situations, sometimes the interval is too narrow -- e.g., if an expert is overconfident. Sometimes, the interval is too wide -- if the expert is too cautious. …

Why Moving Fast And Breaking Things Makes Sense?, Francisco Zapata, Eric Smith, Vladik Kreinovich

Departmental Technical Reports (CS)

In the traditional approach to engineering system design, engineers usually come up with several possible designs, each improving on the previous ones. In coming up with these designs, they try their best to make sure that their designs stay within the safety and other constraints, to avoid potential catastrophic crashes. The need for these safety constraints makes this design process reasonably slow. Software engineering at first followed the same pattern, but then realized that since in most cases, failure of a software test does not lead to a catastrophe, it is much faster to first ignore constraints and then adjust …

Why 70/100 Is Satisfactory? Why Five Letter Grades? Why Other Academic Conventions?, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Why 70/100 is usually a threshold for a student's satisfactory performance? Why there are usually only five letter grades? Why the usual arrangement of research, teaching, and service is 40-40-20? We show that all these arrangements -- and other similar academic arrangements -- can be explained by two ideas: the Laplace Indeterminacy Principle and the seven plus minus two law.

Blessings, God, Sacrifices: Possible Rational Explanations Of Biblical Ideas, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show that many seemingly irrational Biblical ideas can actually be rationally interpreted: that God is everywhere, that we can only say what God is not, that God's name is holy, why cannot you bless as many people as you want, etc. We do not insist on our interpretations, there probably are many others, our sole objective was to show that many Biblical ideas can be rationally explained.