UNIVERSITY OF NORTH FLORIDA & MAYO CLINIC-JAX
.01

ABOUT

PERSONAL DETAILS
1 UNF DR. DEPT. OF MATHEMATICS & STATISTICS, JACKSONVILLE, FL 32224
mapiconimg
mdedeo@unf.edu
904-620-2653
Welcome! My research involves Data Science, Computational Science and Number Theory. Current projects are analysis of the DEA's ARCOS database in NE Florida, 'Refusal of Treatment and Survivability of Patients with Glioblastoma' with the Mayo Clinic - Jacksonville, 'Detection of pediatric pre-ictal events using spectral frequency analysis', 'Search for OPNs: A Computational Sandbox', and more. Click the cloud on the top left of this page to download my CV.

BIO

ABOUT ME

Graduated with a Ph.D. in Mathematics from the University of California at San Diego in 1998 on theoretical models of efficient communication networks. My adviser Audrey Terras is a wonderful mathematician and friend. I am also grateful for all of the help from Harold Stark and Ron Evans. Fan Chung Graham is also an influence of mine as well as Winnie Li.

HOBBIES

INTERESTS

I have river rafted in the US and Venezuela. I'm an advanced certified PADI scuba diver and have done dives in Hawaii and through most of the Caribbean. Over the years I have visited 24 countries on 3 continents, but I'm still waiting to visit Asia, Africa, Australia and Antarctica!

FACTS

FACTS ABOUT ME

Besides a love of computational science, I have a love of art and music. I have painted, made ceramics including raku and more recently have learned glass blowing. I play flute and oboe. Many moons ago I was first chair oboe in the Boulder Youth Symphony, then played jazz flute my first year in college. I truly believe that mathematics and art are one: they both are fascinating in their evolution. I also love the outdoors: snow skiing, scuba diving and camping. I've been a Scout leader for over 10 years, have been to 3 of the four high adventure bases (No. Tier, SeaBase & Summit) and have been on staff for 3 major events (National Jamboree, NOAC and World Jamboree).
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CURRENT CLASSES

FALL 2022
Projects number 2

FINITE MATHEMATICS

FINITE MATHEMATICS

About The Project
Please login to Canvas for the updated syllabus and due dates for assessments! If you have any questions or would like to schedule office hours, please email via Canvas or to mdedeo@unf.edu.

COMPLEX ANALYSIS

COMPLEX ANALYSIS

About The Project
Please login to Canvas for the updated syllabus and due dates for assessments! If you have any questions, please post them to Piazza or email mdedeo@unf.edu.
Mandelbrot set
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POSITIONS

CURRENT
  • NOW
    2017
    JACKSONVILLE

    RESEARCH COLLABORATOR

    MAYO CLINIC - JACKSONVILLE

    Work with physicians, clinicians and organizational personnel on data science and biostatistical projects.
  • NOW
    1998
    JACKSONVILLE

    PROFESSOR (2022-present)

    UNIVERSITY OF NORTH FLORIDA

    Assistant Professor (1998-2005) and Associate Professor (2005-2022) Teach classes of 35 to 140 students ranging from Finite Mathematics to Graduate Number Theory and Complex Analysis.; Incorporate technology in the classroom; Active on multiple departmental and university-wide committees.
TEACHING HISTORY
  • 6/1998
    1/1997
    SAN DIEGO

    LECTURER

    UNIVERSITY OF SAN DIEGO

    Taught one class each semester of either Harvard or Multivariate calculus to classes of 30 students.
  • 6/1997
    1/1994
    SAN DIEGO

    ADJUNCT LECTURER

    SAN DIEGO CITY COLLEGE

    Taught two classes per semester (one in summer) to classes of 30 to 50 students; Class topics ranged from Intermediate Algebra to Calculus II and Introduction to Statistics.
  • 1998
    1993
    SAN DIEGO

    TEACHING ASSISTANT

    UNIVERSITY OF CALIFORNIA - SAN DIEGO

    Assisted every quarter classes of 50 to 160 students. Classes ranged from Harvard and Vector Calculus to Complex Analysis and ODE. Used TI-83 and TI-85 calculators as well as Mathematica and Matlab.
  • 6/1993
    1/1992
    LOS ANGELES

    TEACHING ASSOCIATE

    CALIFORNIA STATE UNIVERSITY - LOS ANGELES

    Taught two classes per quarter of 30 to 50 students. Classes included Math for Elementary Teachers and College Algebra. Also taught at-risk freshmen mathematics and college skills during summer bridge.
.04

RESEARCH

RESEARCH TEAM

JOSEPH FREE

COLLABORATOR

MATTHEW THOMAS

COLLABORATOR

Rukaiyah Husain

Research Assistant

RESEARCH PROJECTS

Socioeconomic Effects of the Opioid Crisis in Northeast Florida

Determining relationships between the socioeconomic variables and opioids in Northeast Florida

Identifying statistical relationships between the opioid epidemic and socioeconomic variables such as unemployment, crime, overdose deaths, education and race in Northeast Florida using latent class analysis, time series analysis and clustering algorithms on the longitudinal data recently made available in the DEA’s ARCOS (Drug Enforcement Agency’s Automated Reports and Consolidated Ordering System) database.

In Search of Odd Perfect Numbers (OPNs)

Computational schemes and difficulties in the identification and existence of OPNs

Although many algorithmic approaches to finding odd perfect numbers (OPNs) have been attempted, this paper presents a novel open-access computational sandbox to search for odd perfect numbers for the purpose of making observations that could lead to new ways to search for OPN’s and their properties.

Refusal trends of recommended therapies for Glioblastoma Multiforme

Identifing the trends of refusal of recommended front line Anti-GBM treatments and its impact on the overall survival rates

With any cancer diagnosis, patients may refuse all or parts of the recommended treatment. Using the NCDB national cancer database for patients with newly diagnosed GBM between 2004 ans 2013, this study focuses on assessing the patterns of denying recommended treatments and its impact on the outcomes.
PAST PROJECTS

Florida Data Science for Social Good (FL-DSSG)

PROJECT LEAD

Work with FL-DSSG program directors to assist interns in solving problems related to data science and good social projects. Provide subject-matter expertise on project tasks and engagement with community partners including Center for Children's Rights – Analyzing Impacts of the School to Prison Pipeline Program; Episcopal Children’s Services – Identifying Children and Families with Low-incomes and Early Learning Needs; and Literacy Alliance of Northeast Florida – Mining Insights from the Adult Learners Educational Data; Barnabas Center – Analyzing Impacts of Health Services on Emergency Department Utilization; League of Women Voters of Florida – Finding Insights to Improve Voter Participation Details at dssg.unf.edu/2020program.html and https://dssg.unf.edu/2021program.html

Detection of pediatric preictal events using spectral frequency analysis

Expansion of power spectral density (PSD) and event-related potential (ERP) analysis in conjunction with visualization techniques

Spectral frequency analysis and visualization techniques are used on pediatric EEG recordings to identify a method of detecting seizures thirty seconds before seizure onset by identifying significant preictal locations and their respective frequencies within the gamma band (30 through 100 Hz).
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AWARDS & EDUCATION

GRANTS SUBMITTED
  • HTTPS://GITHUB.COM/ACCESSARCOS

    $300K NIH R21 GRANT - PI - IN REVISION

    NATIONAL INSTITUTES OF HEALTH

    ACCESS ARCOS: Establishing A Data Repository of Controlled Substance Distribution in the US. Our goal is to create a FAIR, open-source ARCOS data repository and accompanying website (www.accessarcos.com) that provides downloadable ARCOS data in common formats, lists issues we have found in the original data as well as issues that explain differences between different ARCOS reports. Applied for October 15, 2021.
  • $1M NSF ADVANCE GRANT - CO-PI

    NATIONAL SCIENCE FOUNDATION

    UNF's NSF ADVANCE proposal uses intersectional approaches to design systemic change strategies that recognize that gender, race and ethnicity do not exist in isolation from each other and other categories of social identity. In particular, our grant application seeks to create an advancement community for Women and Minorities in STEM at UNF> Applied for November 4, 2021 with A. Schonning (PI), Elizabeth Brown, Amanda Pascale & Susan Perez.
GRANTS AND AWARDS
  • PRESENT
    2019
    JACKSONVILLE

    $1.6M TEACHER QUALITY PARTNERSHIP GRANT (TQP) - CO-PI

    US DEPARTMENT OF EDUCATION

    The Teacher Quality Partnership (TQP) Grant Program aims to increase student achievement by improving the quality of new prospective teachers by enhancing the preparation of prospective teachers and the professional development activities for current teachers; holding teacher preparation programs at institutions of higher education accountable for preparing talented, certified or licensed and effective teachers; and recruiting effective individuals, including minorities and individuals from other occupations, into the teaching force.
  • JACKSONVILLE

    2021 STARS & SOARS Awards

    UNIVERSITY OF NORTH FLORIDA

    Faculty STARS Award for "Effects of Florida’s “Pill Mill” Laws on Opioid Overdoses in NE Florida" & Undergraduate SOARS Project of Merit "Opioid Epidemic on the First Coast" with Jeremiah Baclig, Noah DeDeo, Rukhaiya Husain & Iliya Kulbaka
  • 2021
    2020
    JACKSONVILLE

    DEANS FELLOWSHIP

    UNIVERSITY OF NORTH FLORIDA

    In support of research:
    The Socioeconomic Effect of the Opioid Crisis in Northeast Florida
  • 2018
    JACKSONVILLE

    OUSTANDING UNDERGRADUATE TEACHING AWARD

    UNIVERSITY OF NORTH FLORIDA

    Annual award given to a select number of faculty chosen by their peers for excellence in teaching.
  • JACKSONVILLE

    PROFESSIONAL DEVELOPMENT GRANTS

    UNIVERSITY OF NORTH FLORIDA

    2016 UNF Teaching Grant Recipient:
    Discrete Mathematics II
    2004 UNF Research Grant Recipient:
    Research on Symplectic Maps and Generalizations of the Toda Lattice
    2003 UNF Teaching Grant Recipient:
    Computational Number Theory
    2002 UNF Research Grant Recipient:
    Research on Radon Transforms over k-dimensional rings
  • PAST TEACHING & GRADUATE AWARDS

    1997-98 American Assn. of Univ. Women: American Dissertation Fellow
    1996-97 Univ. of California-San Diego: San Diego Fellowship
    1993-95 SDCC Honor Society: Excellence in Teaching Award
    1991-93 California State Univ.-Los Angeles: Graduate Fellowship
EDUCATION
  • 1998
    1993
    SAN DIEGO

    MATHEMATICS (COMPUTATIONAL SCIENCE) - PHD

    UNIVERSITY OF CALIFORNIA - SAN DIEGO

    Thesis: Graphs over the Ring of Integers Modulo 2^r
  • 1996
    1993
    SAN DIEGO

    APPLIED MATHEMATICS - MA

    UNIVERSITY OF CALIFORNIA - SAN DIEGO

    Computational Science
  • 1993
    1991
    LOS ANGELES

    GENERAL MATHEMATICS - MS

    CALIFORNIA STATE UNIVERSITY - LOS ANGELES

    Thesis: Analytic Functions on the Annulus
  • 1991
    1987
    LOS ANGELES

    MATHEMATICS (HONORS)- BS

    CALIFORNIA STATE UNIVERSITY - LOS ANGELES

.06

SELECT PUBLICATIONS

PUBLICATIONS LIST
14 Mar 2022

Population and Nest Site Evidence for Diamondback Terrapins, Malaclemys terrapin, in Northeast Florida

Frontiers in Ecology and Evolution

Diamondback terrapins (Malaclemys terrapin) are listed as Vulnerable by the IUCN Red List Index of Threatened Species. We conducted headcounts and performed land surveys of shorelines and high spots for evidence of terrapin presence. During the land surveys, we searched for crawls, intact and depredated nests, dead terrapins, and terrapin bones. To evaluate whether woody plant presence affected nest site choices, we recorded the occurrence of 10 common woody plant species during each land survey and compared areas where nesting did and did not occur. DOI: 10.3389/fevo.2022.833199

J. Butler, M. DeDeo, D. Lambert, and D. Murphy.

Population and Nest Site Evidence for Diamondback Terrapins, Malaclemys terrapin, in Northeast Florida

J. Butler, M. DeDeo, D. Lambert, and D. Murphy.
About The Publication
Among the challenges terrapins encounter are habitat loss due to coastal development and sea-level rise, mortality at all life stages by mammalian and avian predators, road mortality, boat strikes, harvest for the pet trade, and drowning in crab traps. The primary objective of this study was to locate populations and nesting areas of diamondback terrapins in the four northeastern-most counties of Florida (Nassau, Duval, St. Johns, and Flagler). We collected 404 records of terrapin activity in 2013 and 2014. Most were from Nassau County (277) and only one was from Flagler County. Most data were in the form of depredated nests (205) and terrapin remains (147). The woody plant data suggest that terrapins were significantly more likely to nest when Christmas berry (Lycium carolinianum) was present, and nesting was less likely when either wax myrtle (Myrica cerifera) or oak (Quercus spp.) were present.
23 Feb 2022

Demographics and surgery-related complications lead to 30-day readmission rates among knee arthroscopic procedures

nee Surgery, Sports Traumatology, Arthroscopy

The study objectives were (1) to evaluate risk factors related to 30-day hospital readmissions after arthroscopic knee surgeries and (2) to determine the complications that may arise from surgery. DOI: 10.1007/s00167-022-06919-2. PMID: 35199185

C Williams, M Bagwell, M DeDeo, A Baker Lutz, M J Deal, B Richey, I Zeini, B Service, D H Youmans, D Osbahr

Demographics and surgery-related complications lead to 30-day readmission rates among knee arthroscopic procedures

C Williams, M Bagwell, M DeDeo, A Baker Lutz, M J Deal, B Richey, I Zeini, B Service, D H Youmans, D Osbahr
About The Publication
There were 83,083 knee arthroscopic procedures between 2012 and 2017 obtained from the National Surgical Quality Improvement Program database. The overall readmission rate was 0.87%. The complication rates were highest for synovectomy and cartilage procedures, 1.6% and 1.3% respectively. A majority of readmissions were related to the procedure (71.1%) with wound complications being the primary reason (28.2%) followed by pulmonary embolism and deep vein thrombosis, 12.7% and 10.6%, respectively. Gender and body mass index were not significant factors and age over 65 years was an independent risk factor. Wound infection, deep vein thrombosis, and pulmonary embolism were the most prevalent complications.
01 Jun 2021

Refusal trends of recommended therapies for Glioblastoma Multiforme | Submitted

Cancer

Glioblastoma multiforme (GBM) is a fatal primary malignancy of the brain with a five-year survival rate of <10%. In this study, we aimed to identify the trends of refusal of recommended front-line Anti-GBM treatments and its impact on the overall survival rates.

Journal Paper Sonikpreet Aulakh, Michelle DeDeo, Naomi Fei, Sanjay Bhatia, Steven Rosenfeld, Aneel Paulus, Alak Manna, Asher Chanan-Khan, Sikander Ailawadhi

Refusal trends of recommended therapies for Glioblastoma Multiforme | Submitted

Sonikpreet Aulakh, Michelle DeDeo, Naomi Fei, Sanjay Bhatia, Steven Rosenfeld, Aneel Paulus, Alak Manna, Asher Chanan-Khan, Sikander Ailawadhi
Journal Paper
About The Publication
Glioblastoma multiforme (GBM) is the most common primary malignant tumor of the brain with a dismal prognosis. Approximately 30.9% of these primary brain tumors are malignant, with GBM being the most commonly identified malignant histology with an annual-age adjusted incidence rate of 14.7%. The annual age-adjusted incidence for GBM among persons age above 40 years is 3.21 per 100,000 population. Further, lack of effective therapies lead to inevitable relapse and approximately 70% of the patients die with in first 2 years of diagnosis. Of all the malignant brain tumors, GBM has the worst prognosis with five-year survival of 5.6%. In majority of cases, treatment recommendations are based on the patient’s current performance status, molecular profile and sub-type of GBM. Yet, patients may refuse all or parts of the recommended treatment. In malignancies including head and neck, lung, and pancreatic cancers – subsets that refused treatments have been identified to be greater than 80 years old, of African American descent, carrying non-private insurance, and presence of increased number of comorbidities as predictors of refusing recommended therapies. Using a national database for patients with newly diagnosed GBM, this study is focusing on assessing the patterns of denying recommended treatments and its impact on the outcomes.
09 Jun 2021

Early detection of pediatric seizures in the high gamma band

IEEE Access

Extensive data analysis uses an expansion of power spectral density analysis and event-related potential analysis in conjunction with visualization techniques to reveal a method of predicting seizures before seizure onset by identifying significant preictal locations in the brain and their respective frequencies within the gamma band (70 through 100 Hz) by utilizing MIT’s Physionet scalp EEG database which consisted of data from acquired from pediatric patients at the Children’s Hospital Boston. DOI: 10.1109/ACCESS.2021.3087782

Journal Paper Michelle DeDeo and Maanasi Garg

Early detection of pediatric seizures in the high gamma band

Michelle DeDeo and Maanasi Garg
Journal Paper
About The Publication
Using spectral frequency analysis and visualization techniques on pediatric electroencephalogram recordings, we identify a method of detecting seizures thirty seconds before seizure onset by identifying significant preictal locations and their respective frequencies within the gamma band of 30 through 100 Hz. Using average log-power differences and visualization techniques to identify significant preictal patterns, intractable seizures are found to have common frequency extremes (CFEs) in the high gamma band between 70 and 100 Hz. Using this data, machine learning detection algorithm predictive performance may be improved by incorporating high gamma band signal processing at the varying location(s) and strength of a pediatric patient’s CFEs. DOI: 10.1109/ACCESS.2021.3087782
21 Jul 2020

The Heat Equation on the Finite Poincare Upper Half Plane

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY

A differential-difference operator is used to model the heat equation on a finite graph analogue of Poincare’s upper half-plane. Finite analogues of the classical theta functions are shown to be solutions to the heat equation in this setting.
DOI: 10.1090/proc/15610

Journal Paper Selected M.R. DeDeo and Elinor Velazquez

The Heat Equation on the Finite Poincare Upper Half Plane

M.R. DeDeo and Elinor Velazquez
Journal Paper Selected
About The Publication
Classically, the heat equation on bounded domains, specifically on tori, yield theta functions as solutions. We study a finite analogue of the Poincare upper half-plane, namely, the finite upper half-plane introduced by Terras. Using this case, we investigate the periodicity inherent in the solutions of bounded domains. The solutions involve zonal spherical functions which come with a natural periodicity. In addition, the related theta functions are automorphic forms. The resultant periodicity interweaves representation theory with the heat equation. We hope this paper stimulates more study of this interplay. DOI: 10.1090/proc/15610
Proceedings of the American Mathematical Society
10 May 2020

On the Energy of Transposition Graphs | Accepted

Combinatorics, Graph Theory & Computing

We analyze and compare properties of Cayley graphs of permutation graphs called transposition graphs as this family of graphs has better degree and diameter properties than other families of graphs. Cayley graphs are directly related to the properties of its generator set and thus Cayley graphs of permutation groups generated by transpositions inherit almost all of the properties of the hypercube. In particular, we study properties of the complete transportation, (transposition) star graph, bubble-sort graph, modified bubble-sort graph and the binary hyper-cube and use these properties to determine bounds on the energy of these graph.

Journal Paper M.R. DeDeo

On the Energy of Transposition Graphs | Accepted

M.R. DeDeo
Journal Paper
About The Publication
Here, we analyze and compare properties of Cayley graphs of permutation graphs called transposition graphs as this family of graphs has better degree and diameter properties than the hypercube. First, the conjugacy class of the permutation group combined with its generators dictates the type of symmetry possessed by their respective Cayley graphs. In addition, the base-b hypercube of dimension n is a class of networks known to possess virtually every known notion of symmetry. Second, properties of the transposition graphs are given. Third, the spectrum of this class of graphs is discussed and formulas for the energy of these graphs are given as they are dependent on these properties.
01 Jun 2019

A Graph Energy Upper Bound Using Spectral Moments

CONGRESSUS NUMERANTIUM

An upper bounds on the energy of graphs is obtained using the spectral moments of the eigenvalues of adjacency matrix associated to the graph using the method of Lagrange multipliers and properties of cubics.

Journal Paper M.R. DeDeo

A Graph Energy Upper Bound Using Spectral Moments

M.R. DeDeo
Journal Paper
About The Publication
In this paper, we find another bound on the energy of all graphs including bipartite graphs using spectral moments and properties of cubics. The goal is to find bounds on the energy E(G) of graphs by using the second and fourth spectral moments, M2 and M4 using Lagrange multipliers.
24 May 2019

Survival trends in glioblastoma and association with treating facility volume

Journal of Clinical Neuroscience

Glioblastoma (GBM) is one of the most lethal cancers. To further understand this extremely poor prognosis disease, we evaluated the effect of the treatment facility volumes on overall survival (OS) on a total of 60,672 eligible GBM patients with median age of 65 years over the years 2004 through 2013.
DOI: 10.1016/j.jocn.2019.04.028

Journal Paper Selected Sonikpreet Aulakh, Michelle DeDeo, Joseph Free, Steven Rosenfeld, Alfredo Quinones-Hinojosa, Aneel Paulus, Alak Manna, Rami Manochakian, Asher Chanan-Khan, Sikander Ailawadhi

Survival trends in glioblastoma and association with treating facility volume

Sonikpreet Aulakh, Michelle DeDeo, Joseph Free, Steven Rosenfeld, Alfredo Quinones-Hinojosa, Aneel Paulus, Alak Manna, Rami Manochakian, Asher Chanan-Khan, Sikander Ailawadhi
Journal Paper Selected
About The Publication
To further understand this extremely poor prognosis disease, we evaluated the effect of the treatment facility volumes on overall survival (OS) over the years, especially after the approval of multimodality therapy using temozolomide (TMZ) in 2005. National Cancer Data Base (NCDB) was utilized to identify GBM cases from 2004 through 2013 using ICD-O-3 code 9440/3 to identify eligible patients. We focused on studying the association between treatment facility volume and OS after adjusting for the patient-, disease-, and facility-characteristics. A total of 60,672 eligible GBM patients with median age of 65 years, treated at 1166 facilities were included in this analysis. The median annual facility volume was 3 patients/year (range: 0.1–55.1) and median OS was 8.1 months. There was an improvement in OS across all facilities after 2005, when multimodality therapy with TMZ was approved. Treatment at quartile 4 centers (Q4; >7 patients/year) was independently associated with decreased all-cause mortality in a multivariate analysis (Q3 hazard ratio [HR]: 1.11, 95% CI 1.09, 1.13; Q2 HR: 1.15, 95% CI 1.12, 1.19; Q1 HR: 1.25, 95% CI 1.17, 1.33). Treatment facility volume independently affects OS among GBM patients. Factors that are variable in high- and low-volume centers should be addressed to mitigate outcome disparities. with Sonikpreet Aulakh, MD, Joseph Free, Aneel Paulus MD, Steven Rosenfeld MD,PhD, Alfredo Quinones-Hinojosa MD, Alak Manna PhD, Rami Manochakian MD, Asher Chanan-Khan MD, J Clin Neurosci. 2019 Oct; 68:271-274. doi: 10.1016/j.jocn.2019.04.028
01 Jun 2007

On the Ramanujancy of Heisenberg graphs over rings of degree 6 or more

CONGRESSUS NUMERANTIUM

Proving conditions for Ramanujancy and non-Ramanujancy is desirable as a Ramanujan graph can represent efficient communication networks in that they minimize cost of wiring and maintenance, but maximize the number of connections from one vertex to another. Here we determine if certain families of Heisenberg graphs satisfy the Ramanujan bound.

Journal Paper M.R. DeDeo, Vincent Dang and Yang Ge

On the Ramanujancy of Heisenberg graphs over rings of degree 6 or more

M.R. DeDeo, Vincent Dang and Yang Ge
Journal Paper
About The Publication
This paper determines if certain families of Heisenberg graphs satisfy the Ramanujan bound. Proving conditions for Ramanujancy and non-Ramanujancy is desirable as a Ramanujan graph can represent efficient communication networks in that they minimize cost of wiring and maintenance, but maximize the number of connections from one vertex to another. We say that a graph is Ramanujan if it is k-regular, simple, connected, undirected and the largest nontrivial eigenvalue, λ, of its adjacency matrix satisfies the condition that |λ|≤sqrt(2k-1) where k is the degree of the graph. We find λ by using the exponential sum lemma and prove a general theorem for the non-Ramanujancy of these graphs under certain conditions. One of these conditions is satisfied by assuming the minimality of certain terms in the exponential sum. Then, without loss of generality, we isolate and prove the conditions for the occurrence of all lower-dimensional eigenvalues that exceed the Ramanujan bound. Also, graphical analysis of the spectra of Ramanujan graphs is performed and it is shown that, for a fixed number of elements in the symmetric set and as a prime p increases without bound, the distribution of the eigenvalues resembles the Sato-Tate semi-circle distribution. with Vincent Dang and Yang Ge, Proceedings of the Thirty-Eighth Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 185 (2007), 111–126. MR2408803 05C50 On the Ramanujancy of Heisenberg Graphs of degree 6 or more
06 May 2007

The Spectrum of Platonic Graphs over Finite Fields

DISCRETE MATHEMATICS

We give a decomposition theorem for Platonic graphs over finite fields and use this to determine the spectrum of these graphs. We also derive estimates for the isoperimetric numbers of the graphs.
DOI: 10.1016/j.disc.2006.07.032

Journal Paper M. DeDeo, Dominic Lanphier and Marvin Minei

The Spectrum of Platonic Graphs over Finite Fields

M. DeDeo, Dominic Lanphier and Marvin Minei
Journal Paper
About The Publication
For q = p, the graph G(q) is called the qth Platonic graph. The name comes from the fact that when p = 3 or 5, the graphs G(p) correspond to 1-skeletons of two of the Platonic solids, in other words, the tetrahedron and the icosahedron as in [2]. More generally, G(p) is the 1-skeleton of a triangulation of the modular curve X(p). In this paper, we determine the spectrum of G(q) and thus generalize results of P.E. Gunnells, “Some elementary Ramanujan graphs”. with Dominic Lanphier and Marvin Minei, Discrete Math. 307 (2007), no. 9-10, 1074–1081.  DOI: 10.1016/j.disc.2006.07.032   MR2292536 05C25 (05C50) 
25 Aug 2005

The Radon transform on the ring of Integers modulo n

SIAM JOURNAL OF DISCRETE MATHEMATICS

The Radon transform on Zkn averages a function over its values on a translate of a fixed subset S in Zkn. We discuss invertibility conditions and computer inverse formulas based on the Moore–Penrose inverse and on linear algorithms. We expect the results to be of use in directional and toroidal time series.
DOI: 10.1137/S0895480103430764

Journal Paper M.R. DeDeo and Elinor Velazquez

The Radon transform on the ring of Integers modulo n

M.R. DeDeo and Elinor Velazquez
Journal Paper
About The Publication
This paper continues the authors previous work that deals with the problem of reconstructing some function defined on a finite group G from averages of the function over various neighborhoods of points. It generalizes work done earlier by Diaconis and Graham, and Fill, among others. Diaconis and Graham discuss the cases where G = Zk2, the group of binary k-tuples, and where G = Sn, the symmetric group on n letters, in order to provide an exposition on discrete Radon transforms which appear in applied statistics. Fill examines the case G = Zk2, the group of integers modulo n. This case arises in directional data analysis and circular time series. Other finite analogues of the Radon transform occur, but we shall restrict ourselves to the case G = Zkn, the group of k-tuples of the integers modulo n, as these results are of use in k-dimensional toroidal time series. with Elinor Velasquez. SIAM J. Discrete Math. 18 (2004/05), no. 3, 472–478. MR2134409 44A12  (43A75) DOI: 10.1137/S0895480103430764
01 Jun 2005

Zeta Functions of Heisenberg Graphs over Finite Rings

Theory and Applications of Special Functions

We investigate Ihara-Selberg zeta functions of Cayley graphs for the Heisenberg group over finite rings Z/pnZ, where p is a prime. In order to do this, we must compute the Galois group of the covering obtained by reducing coordinates in Z/pn+1Z modulo pn.
DOI: 10.1007/0-387-24233-3_8

Chapter Michelle DeDeo, María Martínez, Archie Medrano, Marvin Minei,Harold Stark,Audrey Terras

Zeta Functions of Heisenberg Graphs over Finite Rings

Michelle DeDeo, María Martínez, Archie Medrano, Marvin Minei,Harold Stark,Audrey Terras
Chapter
About The Publication
The aim of this paper is to study the special functions known as Ihara Selberg zeta functions for Cayley graphs of finite Heisenberg groups as well as their factorizations into products of Artin-Ihara L-functions. The Heisenberg group H(R) over a ring R consists of upper triangular 3×3 matrices with entries in R and ones on the diagonal. The Ihara-Selberg zeta function is analogous to the Riemann zeta function with primes replaced by certain closed paths in a graph. This paper is a continuation of (DeDeo et al., 2004) where we presented a study of the statistics of the spectra of adjacency matrices of finite Heisenberg graphs. with Martínez M., Medrano A., Minei M., Stark H., Terras A. (2005) Zeta Functions of Heisenberg Graphs over Finite Rings. In: Ismail DOI: 10.1007/0-387-24233-3_8     MR2132463     05C25 (05C38 11M41 11R42 20H25) 
12 May 2004

An Introduction to Symplectic Maps and Generalizations of the Toda Lattice

CONGRESSUS NUMERANTIUM

We construct particular Hamiltonian systems associated to graphs embedded in Euclidean n−dimensional space and apply the dual transformation to this system. This result extends the one-dimensional dual transformation associated to the Toda lattice. We also discuss some applications to solid-state lattice theory.

Journal Paper M.R. DeDeo and Elinor Velazquez

An Introduction to Symplectic Maps and Generalizations of the Toda Lattice

M.R. DeDeo and Elinor Velazquez
Journal Paper
About The Publication
In this paper, we study the dual transformation applied to N particles and N −1 springs that are no longer constrained to the real line. For simplicity, assume the Hamiltonian system formed to be embedded in Euclidean n−space with the standard metric and also assume that N < ∞ and that all N particles are connected to at least one spring. Our interpretation of one possible extension is to create graphs that can act as dynamical systems. We decide to take a graph (a finite one, for simplicity), position masses at the vertices of the graph, and exchange the edges of the graphs with springs. Again, for simplicity, we assume that the springs behave uniformly, although not necessarily linearly. We then re-express the Hamiltonian with respect to the graphs rather than focusing on the underlying manifold. We focus on graphs which may be neatly embedded into Euclidean n−dimensional space for the purpose of this paper. Hence we now have information regarding how graphs behave when “equipped” with a symplectic 2−form.   with Elinor Velasquez. Proceedings of the Thirty-Fifth Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congressus Numerantium 169 (2004), 127–139.   MR2122063     70H06 (05C45 37J35 37K60 70H15 82C05) An Introduction to Symplectic Maps and Generalizations of the Toda Lattice  
10 May 2003

Generalized Kloosterman sums over Rings of Order 2r

CONGRESSUS NUMERANTIUM

After presenting several applications of Kloosterman sums in analytic number theory, generalized Kloosterman sums attached to Gauss sums over the ring of 2r elements are explicitly evaluated and are shown to be sines and cosines

Journal Paper M. DeDeo

Generalized Kloosterman sums over Rings of Order 2r

M. DeDeo
Journal Paper
About The Publication
Although Kloosterman sums of odd prime powers have been studied extensively, many authors avoid the case where p = 2. This case is important as Kloosterman sums for fields of characteristic 2 have been useful in error-correcting codes and Kloosterman sums over the ring of integers modulo pr have been helpful in studying the spectra of Euclidean graphs. After stating some well-known facts about Kloosterman sums, we evaluate several generalized Kloosterman sums modulo 2r. Proceedings of the Thirty-Fourth Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 165 (2003), 65–75MR2049122 11T24 (11L05) Generalized Kloosterman sums over Rings of Order 2r
01 Jun 2003

Spectra of Heisenberg Graphs over Finite Rings

Dynamical systems and differential equations

We investigate spectra of Cayley graphs for the Heisenberg group over finite rings Z/pnZ, where p is a prime. Emphasis is on graphs of degree four. We show that for odd p there is only one such connected graph up to isomorphism. When p = 2, there are at most two isomorphism classes. We study the spectra using representations of the Heisenberg group. This allows us to produce histograms and butterfly diagrams of the spectra. DOI: 10.3934/proc.2003.2003.213

Journal Paper M. DeDeo, M. Martinez, A. Medrano, M. Minei, H.M. Stark, and A. Terras

Spectra of Heisenberg Graphs over Finite Rings

M. DeDeo, M. Martinez, A. Medrano, M. Minei, H.M. Stark, and A. Terras
Journal Paper
About The Publication
The aim of this paper is to study spectra of Cayley graphs attached to finite Heisenberg groups using histograms and Hofstadter butterfly figures. The Heisenberg group H(R) over a ring R consists of upper triangular 3×3 matrices with entries in R and ones on the diagonal. Heisenberg groups over finite fields have provided a tool in the search for random number generators (see Maria Zack) as well as the search for Ramanujan graphs (see Perla Myers). Other references are P. Diaconis and L. Saloff-Coste [5] and Terras [24]. As shown in the last reference, the size of the second largest (in absolute value) eigenvalue of the adjacency matrix governs the speed of convergence to uniform for the standard random walk on a connected regular graph. Ramanujan graphs have the best possible eigenvalue bound for connected regular graphs of fixed degree in an infinite sequence of graphs with number of vertices going to infinity. For such graphs, the random walker gets lost as quickly as possible. Equivalently, this says that such graphs can be used to build efficient communication networks. with M. Martinez, A. Medrano, M. Minei, H.M. Stark, and A. TerrasDynamical systems and differential equations (Wilmington, NC, 2002). Discrete Contin. Dyn. Syst. 2003, suppl., 213–222. MR2018119 11T60 (05C25) https://www.aimsciences.org/article/doi/10.3934/proc.2003.2003.213
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MR4032784  Lai, Tim; Marino, Alicia; Robinson, Angela; Wan, Daqing Moment subset sums over finite fields. Finite Fields Appl. 62 (2020), 101607, 25 pp. (Reviewer: Michelle R. DeDeo) 11T24 (11T71)
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MR3996348 Ding, Yuchen; Zhou, Haiyan Subset sums over Galois rings. J. Number Theory 205 (2019), 148–169. (Reviewer: Michelle R. DeDeo) 11T24 (05A15 05A16 11B13)
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MR3964251 Choe, Hyok; Choe, Chunghyok The k-subset sum problem over finite fields of characteristic 2. Finite Fields Appl. 59 (2019), 175–184. (Reviewer: Michelle R. DeDeo) 11T24 (11T71)
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MR3912936 Kalmynin, A. B. Large values of short character sums. J. Number Theory 198 (2019), 200–210. (Reviewer: Michelle R. DeDeo) 11L07
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MR3803730 Tu, Fang-Ting; Yang, Yifan Evaluation of certain hypergeometric functions over finite fields. SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018), Paper No. 050, 18 pp. (Reviewer: Michelle R. DeDeo) 11T23 (11G05 11G30 11T24)
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MR3759792 Momihara, Koji; Xiang, Qing Strongly regular Cayley graphs from partitions of subdifference sets of the Singer difference sets. Finite Fields Appl. 50 (2018), 222–250. (Reviewer: Michelle R. DeDeo) 05E30 (05B10)
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MR3648527 Zhang, Wen Peng; Li, Xiao Xue The fourth power mean of the general 2-dimensional Kloostermann sums modp. Acta Math. Sin. (Engl. Ser.) 33 (2017), no. 6, 861–867. (Reviewer: Michelle R. DeDeo) 11L05
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MR3536006 Lu, Yaming Mean value of Kloosterman sums over short intervals. Chin. Ann. Math. Ser. B 37 (2016), no. 5, 743–752. (Reviewer: Michelle R. DeDeo) 11L05
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MR3440174 Shuman, David I.; Ricaud, Benjamin; Vandergheynst, Pierre Vertex-frequency analysis on graphs. Appl. Comput. Harmon. Anal. 40 (2016), no. 2, 260–291. (Reviewer: Michelle R. DeDeo) 94A12
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MR3354946 Atanasov, Risto; Budden, Mark; Lambert, Joshua Character difference digraphs over finite fields. Rend. Semin. Mat. Univ. Padova 133 (2015), 103–115. (Reviewer: Michelle R. DeDeo) 05C30 (05C20 11T24)
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MR3276346 Lin, Yi-Hsuan; Tu, Fang-Ting Twisted Kloosterman sums. J. Number Theory 147(2015), 666–690. (Reviewer: Michelle R. DeDeo) 11L05
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MR3309823 Ghayekhloo, Somayeh; Erfanian, Ahmad; Tolue, Behnaz The generalised non-commuting graph of a finite group. C. R. Acad. Bulgare Sci. 67 (2014), no. 8, 1037–1044. (Reviewer: Michelle R. DeDeo) 05C25
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MR3223318 Bourgain, J.; Garaev, M. Z. Kloosterman sums in residue rings. Acta Arith. 164(2014), no. 1, 43–64. (Reviewer: Michelle R. DeDeo) 11L05
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MR3185227  Aoki, Noboru On sign ambiguities of multiplicative relations between Gauss sums. Comment. Math. Univ. St. Pauli 62 (2013), no. 2, 143–171. (Reviewer: Michelle R. DeDeo)11L05
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MR3108074 Maeda, Taiki Gauss sums on GL2(Z/plZ). J. Algebra 396 (2013), 98–116. (Reviewer: Michelle R. DeDeo) 11L05
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MR3081777  Demirci Akarsu, Emek; Marklof, Jens The value distribution of incomplete Gauss sums. Mathematika 59 (2013), no. 2, 381–398. (Reviewer: Michelle R. DeDeo) 11L05
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MR3016560 Bassalygo, L. A.; Zinoviev, V. A. On divisibility of exponential sums of polynomials of special type over fields of characteristic 2. Des. Codes Cryptogr. 66 (2013), no. 1-3,129–143. (Reviewer: Michelle R. DeDeo) 11L05 (11T23 12E10)
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MR2944570 Xia, Lingli; Yang, Jing Sign or root of unity ambiguities of certain Gauss sums.Front. Math. China 7 (2012), no. 4, 743–764. (Reviewer: Michelle R. DeDeo) 11L05 (11T24)
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MR2836365  Kopparty, Swastik; Lev, Vsevolod F.; Saraf, Shubhangi; Sudan, Madhu Kakeya-type sets in finite vector spaces. J. Algebraic Combin. 34 (2011), no. 3, 337–355. (Reviewer: Michelle R. DeDeo) 11T30 (05D05 52B55)
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MR2813459 Zhang, Xiyong; Guo, Hua; Gao, Zongsheng Characterizations of bent and almost bent function on Z2p. Graphs Combin. 27 (2011), no. 4, 603–620. (Reviewer: Michelle R. DeDeo) 11L05 (05B15 11T71)
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MR2718845 Yang, Jing; Xia, LingLi Complete solving of explicit evaluation of Gauss sums in the index 2 case. Sci. China Math. 53 (2010), no. 9, 2525–2542. (Reviewer: Michelle R. DeDeo) 11L05 (11T24 11T71)
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MR2661385 Storm, Christopher An infinite family of graphs with the same Ihara zeta function. Electron. J. Combin. 17 (2010), no. 1, Research Paper 82, 8 pp. (Reviewer: Michelle R. DeDeo) 05C38 (05A15)
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MR2601046  Gurak, S. Gauss sums for prime powers in p-adic fields. Acta Arith. 142 (2010),no. 1, 11–39. (Reviewer: Michelle R. DeDeo) 11L05 (11S85)
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MR2592482 Omidi Koma, B.; Panario, D.; Wang, Q. The number of irreducible polynomials of degree n over Fq with given trace and constant terms. Discrete Math. 310 (2010), no. 8, 1282–1292. (Reviewer: Michelle R. DeDeo) 11T06
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MR2537711 Gurak, Stanley J. Kloosterman sums for prime powers in p-adic fields. J. Théor. Nombres Bordeaux 21 (2009), no. 1, 175–201. (Reviewer: Michelle R. DeDeo) 11L05 (11S85)
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MR2516974 He, Yuan; Liao, Qunying On an identity associated with Weil’s estimate and its applications. J. Number Theory 129 (2009), no. 5, 1075–1089. (Reviewer: Michelle R. DeDeo) 11L05
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MR2435063 Moisio, Marko; Ranto, Kalle Elliptic curves and explicit enumeration of irreducible polynomials with two coefficients prescribed. Finite Fields Appl. 14 (2008), no. 3, 798–815. (Reviewer: Michelle R. DeDeo) 11L05 (11G20 11T06 11T23)
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MR2397155 Varbanets, S. P. The norm Kloosterman sums over Z[i]. Analytic and probabilistic methods in number theory/Analiziniai ir tikimybiniai metodai skaičių teorijoje, 225–239, TEV, Vilnius, 2007. (Reviewer: Michelle R. DeDeo) 11L05
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MR2310293 Motohashi, Y. Sums of Kloosterman sums revisited. The Conference on L-Functions, 141–163, World Sci. Publ., Hackensack, NJ, 2007. (Reviewer: Michelle R. DeDeo) 11L05 (11F72)
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MR2275479 Liu, Hua Ning; Zhang, Wen Peng General Kloosterman sums and the difference between an integer and its inverse modulo q. Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 1, 77–82. (Reviewer: Michelle R. DeDeo) 11L05 (11T24)
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MR2257593 Kwak, Jin Ho; Sato, Iwao Zeta functions of line, middle, total graphs of a graph and their coverings. Linear Algebra Appl. 418 (2006), no. 1, 234–256. (Reviewer: Michelle R. DeDeo)05C50 (11M41)
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MR2196576 Zhang, Wenpeng On the fourth power mean of the classical Kloosterman sums.Proceedings of the 16th International Conference of the Jangjeon Mathematical Society, 180–187, Jangjeon Math. Soc., Hapcheon, 2005. (Reviewer: Michelle R. DeDeo) 11L05
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MR1973260 Conder, Marston D. E.; Marušič, Dragan A tetravalent half-arc-transitive graph with non-abelian vertex stabilizer. J. Combin. Theory Ser. B 88 (2003), no. 1, 67–76. (Reviewer: Michelle R. DeDeo) 05C25 (05C38)
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MR1881928 Lucido, Maria Silvia Groups in which the prime graph is a tree. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 5 (2002), no. 1, 131–148. (Reviewer: Michelle R. DeDeo) 05C25 (05C05 20D60)
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MR1849089 Gurak, Stanley Period polynomials for Fp2 of fixed small degree. Finite fields and applications (Augsburg, 1999), 196–207, Springer, Berlin, 2001. (Reviewer: Michelle R. DeDeo) 11T22 (11T24)
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