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Articles 1 - 30 of 96
Full-Text Articles in Mathematics
Topological Comparison Of Some Dimension Reduction Methods Using Persistent Homology On Eeg Data, Eddy Kwessi
Topological Comparison Of Some Dimension Reduction Methods Using Persistent Homology On Eeg Data, Eddy Kwessi
Mathematics Faculty Research
In this paper, we explore how to use topological tools to compare dimension reduction methods. We first make a brief overview of some of the methods often used in dimension reduction such as isometric feature mapping, Laplacian Eigenmaps, fast independent component analysis, kernel ridge regression, and t-distributed stochastic neighbor embedding. We then give a brief overview of some of the topological notions used in topological data analysis, such as barcodes, persistent homology, and Wasserstein distance. Theoretically, when these methods are applied on a data set, they can be interpreted differently. From EEG data embedded into a manifold of high dimension, …
Biofilm Viscoelasticity And Nutrient Source Location Control Biofilm Growth Rate, Migration Rate, And Morphology In Shear Flow, Hoa Nguyen, Abraham Ybarra, H. Başağaoğlu, Orrin Shindell
Biofilm Viscoelasticity And Nutrient Source Location Control Biofilm Growth Rate, Migration Rate, And Morphology In Shear Flow, Hoa Nguyen, Abraham Ybarra, H. Başağaoğlu, Orrin Shindell
Mathematics Faculty Research
We present a numerical model to simulate the growth and deformation of a viscoelastic biofilm in shear flow under different nutrient conditions. The mechanical interaction between the biofilm and the fluid is computed using the Immersed Boundary Method with viscoelastic parameters determined a priori from measurements reported in the literature. Biofilm growth occurs at the biofilm-fluid interface by a stochastic rule that depends on the local nutrient concentration. We compare the growth, migration, and morphology of viscoelastic biofilms with a common relaxation time of 18 min over the range of elastic moduli 10–1000 Pa in different nearby nutrient source configurations. …
Discrete Dynamics Of Dynamic Neural Fields, Eddy Kwessi
Discrete Dynamics Of Dynamic Neural Fields, Eddy Kwessi
Mathematics Faculty Research
Large and small cortexes of the brain are known to contain vast amounts of neurons that interact with one another. They thus form a continuum of active neural networks whose dynamics are yet to be fully understood. One way to model these activities is to use dynamic neural fields which are mathematical models that approximately describe the behavior of these congregations of neurons. These models have been used in neuroinformatics, neuroscience, robotics, and network analysis to understand not only brain functions or brain diseases, but also learning and brain plasticity. In their theoretical forms, they are given as ordinary or …
Analysis Of Eeg Data Using Complex Geometric Structurization, Eddy A. Kwessi, L. J. Edwards
Analysis Of Eeg Data Using Complex Geometric Structurization, Eddy A. Kwessi, L. J. Edwards
Mathematics Faculty Research
Electroencephalogram (EEG) is a common tool used to understand brain activities. The data are typically obtained by placing electrodes at the surface of the scalp and recording the oscillations of currents passing through the electrodes. These oscillations can sometimes lead to various interpretations, depending on, for example, the subject’s health condition, the experiment carried out, the sensitivity of the tools used, or human manipulations. The data obtained over time can be considered a time series. There is evidence in the literature that epilepsy EEG data may be chaotic. Either way, the Embedding Theory in dynamical systems suggests that time series …
Resources For Supporting Mathematics And Data Science Instructors During Covid-19, Eduardo C. Balreira, C. Hawthorne, G. Stadnyk, Z. Teymuroglu, M. Torres, J. R. Wares
Resources For Supporting Mathematics And Data Science Instructors During Covid-19, Eduardo C. Balreira, C. Hawthorne, G. Stadnyk, Z. Teymuroglu, M. Torres, J. R. Wares
Mathematics Faculty Research
In late May of 2020, a few months after the raging COVID-19 pandemic forced university faculty to quickly switch to online teaching, the Associated Colleges of the South (ACS) released a call for grant applications to support working groups "to help faculty within our consortium who will be teaching during the pandemic (e.g., from hybrid courses with some remote/online components to fully remote/online courses; socially distanced face-to-face courses)." We replied to this call and the ACS awarded the six of us (from four ACS schools) a Summer Rapid Response Grant in early June. The grant funded our efforts to create …
On The Equivalence Between Weak Bmo And The Space Of Derivatives Of The Zygmund Class, Eddy Kwessi
On The Equivalence Between Weak Bmo And The Space Of Derivatives Of The Zygmund Class, Eddy Kwessi
Mathematics Faculty Research
In this paper, we will discuss the space of functions of weak bounded mean oscillation. In particular, we will show that this space is the dual space of the special atom space, whose dual space was already known to be the space of derivative of functions (in the sense of distribution) belonging to the Zygmund class of functions. We show, in particular, that this proves that the Hardy space H1 strictly contains the special atom space.
A Consistent Estimator Of Nontrivial Stationary Solutions Of Dynamic Neural Fields, Eddy Kwessi
A Consistent Estimator Of Nontrivial Stationary Solutions Of Dynamic Neural Fields, Eddy Kwessi
Mathematics Faculty Research
Dynamics of neural fields are tools used in neurosciences to understand the activities generated by large ensembles of neurons. They are also used in networks analysis and neuroinformatics in particular to model a continuum of neural networks. They are mathematical models that describe the average behavior of these congregations of neurons, which are often in large numbers, even in small cortexes of the brain. Therefore, change of average activity (potential, connectivity, firing rate, etc.) are described using systems of partial different equations. In their continuous or discrete forms, these systems have a rich array of properties, among which is the …
A Continuous-Time Mathematical Model And Discrete Approximations For The Aggregation Of Β-Amyloid, A. S. Ackleh, Saber Elaydi, G. Livadiotis, A. Veprauskas
A Continuous-Time Mathematical Model And Discrete Approximations For The Aggregation Of Β-Amyloid, A. S. Ackleh, Saber Elaydi, G. Livadiotis, A. Veprauskas
Mathematics Faculty Research
Alzheimer's disease is a degenerative disorder characterized by the loss of synapses and neurons from the brain, as well as the accumulation of amyloid-based neuritic plaques. While it remains a matter of contention whether β-amyloid causes the neurodegeneration, β-amyloid aggregation is associated with the disease progression. Therefore, gaining a clearer understanding of this aggregation may help to better understand the disease. We develop a continuous-time model for β-amyloid aggregation using concepts from chemical kinetics and population dynamics. We show the model conserves mass and establish conditions for the existence and stability of equilibria. We also develop two …
The Special Atom Space And Haar Wavelets In Higher Dimensions, Eddy Kwessi, G. De Souza, N. Djitte, M. Ndiaye
The Special Atom Space And Haar Wavelets In Higher Dimensions, Eddy Kwessi, G. De Souza, N. Djitte, M. Ndiaye
Mathematics Faculty Research
In this note, we will revisit the special atom space introduced in the early 1980s by Geraldo De Souza and Richard O'Neil. In their introductory work and in later additions, the space was mostly studied on the real line. Interesting properties and connections to spaces such as Orlicz, Lipschitz, Lebesgue, and Lorentz spaces made these spaces ripe for exploration in higher dimensions. In this article, we extend this definition to the plane and space and show that almost all the interesting properties such as their Banach structure, Hölder's-type inequalities, and duality are preserved. In particular, dual spaces of special atom …
Discrete Evolutionary Population Models: A New Approach, K. Mokni, Saber Elaydi, M. Ch-Chaoui, A. Eladdadi
Discrete Evolutionary Population Models: A New Approach, K. Mokni, Saber Elaydi, M. Ch-Chaoui, A. Eladdadi
Mathematics Faculty Research
In this paper, we apply a new approach to a special class of discrete time evolution models and establish a solid mathematical foundation to analyse them. We propose new single and multi-species evolutionary competition models using the evolutionary game theory that require a more advanced mathematical theory to handle effectively. A key feature of this new approach is to consider the discrete models as non-autonomous difference equations. Using the powerful tools and results developed in our recent work [E. D'Aniello and S. Elaydi, The structure of ω-limit sets of asymptotically non-autonomous discrete dynamical systems, Discr. Contin. Dyn. Series B. …
Improving Foresight Predictions In The 2002-2018 Nfl Regular-Seasons: A Classic Tale Of Quantity Vs. Quality, Eduardo C. Balreira, Brian K. Miceli
Improving Foresight Predictions In The 2002-2018 Nfl Regular-Seasons: A Classic Tale Of Quantity Vs. Quality, Eduardo C. Balreira, Brian K. Miceli
Mathematics Faculty Research
Utilizing a modied Bradley-Terry model, we develop a method of making foresight predictions of 2002-2018 NFL games by incorporating a home-eld parameter into previously established ranking models. Knowing only the home team and score of each contest, and taking into account previous predictions, we optimize this parameter considering one of two things: the quantity of correct picks to date or the quality of predictions to date as measured by a quadratic scoring
function. Our main results establish that optimization of quality-rather than quantity-when making a prediction has higher overall accuracy.
Effects Of Cell Morphology And Attachment To A Surface On The Hydrodynamic Performance Of Unicellular Choanoflagellates, Hoa Nguyen, M. A. R. Koehl, Christian Oakes, G. Bustamente, L. Fauci
Effects Of Cell Morphology And Attachment To A Surface On The Hydrodynamic Performance Of Unicellular Choanoflagellates, Hoa Nguyen, M. A. R. Koehl, Christian Oakes, G. Bustamente, L. Fauci
Mathematics Faculty Research
Choanoflagellates, eukaryotes that are important predators on bacteria in aquatic ecosystems, are closely related to animals and are used as a model system to study the evolution of animals from protozoan ancestors. The choanoflagellate Salpingoeca rosetta has a complex life cycle with different morphotypes, some unicellular and some multicellular. Here we use computational fluid dynamics to study the hydrodynamics of swimming and feeding by different unicellular stages of S. rosetta: a swimming cell with a collar of prey-capturing microvilli surrounding a single flagellum, a thecate cell attached to a surface and a dispersal-stage cell with a slender body, long …
Effects Of Advective-Diffusive Transport Of Multiple Chemoattractants On Motility Of Engineered Chemosensory Particles In Fluidic Environments, D. King, H. Başağaoğlu, Hoa Nguyen, Frank G. Healy, Melissa Whitman, Sauro Succi
Effects Of Advective-Diffusive Transport Of Multiple Chemoattractants On Motility Of Engineered Chemosensory Particles In Fluidic Environments, D. King, H. Başağaoğlu, Hoa Nguyen, Frank G. Healy, Melissa Whitman, Sauro Succi
Mathematics Faculty Research
Motility behavior of an engineered chemosensory particle (ECP) in fluidic environments is driven by its responses to chemical stimuli. One of the challenges to understanding such behaviors lies in tracking changes in chemical signal gradients of chemoattractants and ECP-fluid dynamics as the fluid is continuously disturbed by ECP motion. To address this challenge, we introduce a new multiscale numerical model to simulate chemotactic swimming of an ECP in confined fluidic environments by accounting for motility-induced disturbances in spatiotemporal chemoattractant distributions. The model accommodates advective-diffusive transport of unmixed chemoattractants, ECP-fluid hydrodynamics at the ECP-fluid interface, and spatiotemporal disturbances in the chemoattractant …
A Discrete Mathematical Model For The Aggregation Of Β-Amyloid, Maher A. Dayeh, George Livadiotis, Saber Elaydi
A Discrete Mathematical Model For The Aggregation Of Β-Amyloid, Maher A. Dayeh, George Livadiotis, Saber Elaydi
Mathematics Faculty Research
Dementia associated with the Alzheimer's disease is thought to be correlated with the conversion of the β − Amyloid (Aβ) peptides from soluble monomers to aggregated oligomers and insoluble fibrils. We present a discrete-time mathematical model for the aggregation of Aβ monomers into oligomers using concepts from chemical kinetics and population dynamics. Conditions for the stability and instability of the equilibria of the model are established. A formula for the number of monomers that is required for producing oligomers is also given. This may provide compound designers a mechanism to inhibit the Aβ aggregation.
A Geometric Generalizaion Of The Planar Gale-Nikaidô Theorem, Eduardo C. Balreira
A Geometric Generalizaion Of The Planar Gale-Nikaidô Theorem, Eduardo C. Balreira
Mathematics Faculty Research
The Gale-Nikaidô Theorem establishes global injectivity of maps defined over rectangular regions provided the Jacobian matrix is a P-matrix. We provide a purely geometric generalization of this result in the plane by showing that if the image of each edge of the rectangular domain is realized as a graph of a function over the appropriate axis, then the map is injective. We also show that the hypothesis that the Jacobian matrix is a P-matrix is simply one way to analytically check this geometric condition.
Global Stability Of Higher Dimensional Monotone Maps, Eduardo C. Balreira, Saber Elaydi, Rafael Luís
Global Stability Of Higher Dimensional Monotone Maps, Eduardo C. Balreira, Saber Elaydi, Rafael Luís
Mathematics Faculty Research
We develop a notion of monotonicity for maps defined on Euclidean spaces Rk+, of arbitrary dimension k. This is a geometric approach that extends the classical notion of planar monotone maps or planar competitive difference equations. For planar maps, we show that our notion and the classical notion of monotonicity are equivalent. In higher dimensions, we establish certain verifiable conditions under which Kolmogorov monotone maps on Rk+ have a globally asymptotically stable fixed point. We apply our results to two competition population models, the Leslie–Gower and the Ricker models of two- and three-species. It …
Generating Functions And Wilf Equivalence For Generalized Interval Embeddings, Russell Chamberlain, Garner Cochran, Sam Ginsburg, Brian K. Miceli, Manda Riehl, Chi Zhang
Generating Functions And Wilf Equivalence For Generalized Interval Embeddings, Russell Chamberlain, Garner Cochran, Sam Ginsburg, Brian K. Miceli, Manda Riehl, Chi Zhang
Mathematics Faculty Research
In 1999 in [J. Difference Equ. Appl. 5, 355–377], Noonan and Zeilberger extended the Goulden-Jackson Cluster Method to find generating functions of word factors. Then in 2009 in [Electron. J. Combin. 16(2), RZZ], Kitaev, Liese, Remmel and Sagan found generating functions for word embeddings and proved several results on Wilf-equivalence in that setting. In this article, the authors focus on generalized interval embeddings, which encapsulate both factors and embeddings, as well as the “space between” these two ideas. The authors present some results in the most general case of interval embeddings. Two special cases of …
Connection Coefficients Between Generalized Rising And Falling Factorial Bases, Jeffrey Liese, Brian K. Miceli, Jeffrey Remmel
Connection Coefficients Between Generalized Rising And Falling Factorial Bases, Jeffrey Liese, Brian K. Miceli, Jeffrey Remmel
Mathematics Faculty Research
Let S = (s1, s2, . . .) be any sequence of nonnegative integers and let Sk = ∑ki=1si. We then define the falling (rising) factorials relative to S by setting (x) ↓k,S=(x - S1)(x - S2) · · · (x - Sk) and (x) ↑k,S= (x + S1)(x + S2) · · · (x + Sk) if k ≥ 1 with (x …
A Combinatorial Proof Of A Theorem Of Katsuura, Brian K. Miceli
A Combinatorial Proof Of A Theorem Of Katsuura, Brian K. Miceli
Mathematics Faculty Research
We give a combinatorial proof of an algebraic result of Katsuura's. Moreover, we use the proof of this result to shed some light on an interesting property of the result itself.
Hydrodynamics Of Diatom Chains And Semiflexible Fibres, Hoa Nguyen, L. Fauci
Hydrodynamics Of Diatom Chains And Semiflexible Fibres, Hoa Nguyen, L. Fauci
Mathematics Faculty Research
Diatoms are non-motile, unicellular phytoplankton that have the ability to form colonies in the form of chains. Depending upon the species of diatoms and the linking structures that hold the cells together, these chains can be quite stiff or very flexible. Recently, the bending rigidities of some species of diatom chains have been quantified. In an effort to understand the role of flexibility in nutrient uptake and aggregate formation, we begin by developing a three-dimensional model of the coupled elastic-hydrodynamic system of a diatom chain moving in an incompressible fluid. We find that simple beam theory does a good job …
Global Dynamics Of Triangular Maps, Eduardo C. Balreira, Saber Elaydi, Rafael Luís
Global Dynamics Of Triangular Maps, Eduardo C. Balreira, Saber Elaydi, Rafael Luís
Mathematics Faculty Research
We consider continuous triangular maps on IN, where I is a compact interval in the Euclidean space R. We show, under some conditions, that the orbit of every point in a triangular map converges to a fixed point if and only if there is no periodic orbit of prime period two. As a consequence we obtain a result on global stability, namely, if there are no periodic orbits of prime period 2 and the triangular map has a unique fixed point, then the fixed point is globally asymptotically stable. We also discuss examples and applications of our …
An Oracle Method To Predict Nfl Games, Eduardo C. Balreira, Brian K. Miceli, Thomas Tegtmeyer
An Oracle Method To Predict Nfl Games, Eduardo C. Balreira, Brian K. Miceli, Thomas Tegtmeyer
Mathematics Faculty Research
Multiple models are discussed for ranking teams in a league and introduce a new model called the Oracle method. This is a Markovian method that can be customized to incorporate multiple team traits into its ranking. Using a foresight prediction of NFL game outcomes for the 2002–2013 seasons, it is shown that the Oracle method correctly picked 64.1% of the games under consideration, which is higher than any of the methods compared, including ESPN Power Rankings, Massey, Colley, and PageRank.
Local Stability Implies Global Stability For The Planar Ricker Competition Model, Eduardo C. Balreira, Saber Elaydi, Rafael Luís
Local Stability Implies Global Stability For The Planar Ricker Competition Model, Eduardo C. Balreira, Saber Elaydi, Rafael Luís
Mathematics Faculty Research
Under certain analytic and geometric assumptions we show that local stability of the coexistence (positive) fixed point of the planar Ricker competition model implies global stability with respect to the interior of the positive quadrant. This result is a confluence of ideas from Dynamical Systems, Geometry, and Topology that provides a framework to the study of global stability for other planar competition models.
The Method Of Lagrange Multipliers, William Trench
The Method Of Lagrange Multipliers, William Trench
Mathematics Faculty Research
This is a supplement to the author's "Introduction to Real Analysis." It has been judged to meet the evaluation criteria set by the Editorial Board of the American Institute of Mathematics in connection with the Institute's Open Textbook Initiative. It may be copied, modified, redistributed, translated, and built upon subject to the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. A complete instructor's solution manual is available by email to wtrench@trinity.edu, subject to verification of the requestor's faculty status.
Functions Defined By Improper Integrals, William Trench
Functions Defined By Improper Integrals, William Trench
Mathematics Faculty Research
This is a supplement to the author's Introduction to Real Analysis. It has been judged to meet the evaluation criteria set by the Editorial Board of the American Institute of Mathematics in connection with the Institute's Textbook Initiative. It may be copied, modified, redistributed, translated, and built upon subject to the Creative Commons license Attribution-NonCommercial-ShareAlike 3.0 Unported License. A complete instructor's solution manual is available by email to wtrench@trinity.edu subject to verification of the requestor's faculty status.
General Allee Effect In Two-Species Population Biology, G Livadiotis, Saber Elaydi
General Allee Effect In Two-Species Population Biology, G Livadiotis, Saber Elaydi
Mathematics Faculty Research
The main objective of this work is to present a general framework for the notion of the strong Allee effect in population models, including competition, mutualistic, and predator–prey models. The study is restricted to the strong Allee effect caused by an inter-specific interaction. The main feature of the strong Allee effect is that the extinction equilibrium is an attractor. We show how a ‘phase space core’ of three or four equilibria is sufficient to describe the essential dynamics of the interaction between two species that are prone to the Allee effect. We will introduce the notion of semistability in planar …
Bifurcation And Invariant Manifolds Of The Logistic Competition Model, M. Guzowska, Rafael Luís, Saber Elaydi
Bifurcation And Invariant Manifolds Of The Logistic Competition Model, M. Guzowska, Rafael Luís, Saber Elaydi
Mathematics Faculty Research
In this paper we study a new logistic competition model. We will investigate stability and bifurcation of the model. In particular, we compute the invariant manifolds, including the important center manifolds, and study their bifurcation. Saddle-node and period doubling bifurcation route to chaos is exhibited via numerical simulations.
A Generalization Of The Fujisawa–Kuh Global Inversion Theorem, M. Radulescu, S. Radulescu, Eduardo C. Balreira
A Generalization Of The Fujisawa–Kuh Global Inversion Theorem, M. Radulescu, S. Radulescu, Eduardo C. Balreira
Mathematics Faculty Research
We discuss the problem of global invertibility of nonlinear maps defined on the finitedimensional Euclidean space via differential tests. We provide a generalization of theFujisawa-Kuh global inversion theorem and introduce a generalized ratio conditionwhich detects when the pre-image of a certain class of linear manifolds is non-emptyand connected. In particular, we provide conditions that also detect global injectivity.
Budding Yeast, Branching Processes, And Generalized Fibonacci Numbers, Peter Olofsson, Ryan C. Daileda
Budding Yeast, Branching Processes, And Generalized Fibonacci Numbers, Peter Olofsson, Ryan C. Daileda
Mathematics Faculty Research
A real-world application of branching processes to a problem in cell biology where the generalized Fibonacci numbers known as k-nacci numbers play a crucial role is described. The k-nacci sequence is used to obtain asymptotics, computational formulas, and to justify certain practical simplifications. Along the way, an explicit formula for the sum of k-nacci numbers is established.
Towards A Theory Of Periodic Difference Equations And Its Application To Population Dynamics, Saber Elaydi, Rafael Luís, Henrique Oliveira
Towards A Theory Of Periodic Difference Equations And Its Application To Population Dynamics, Saber Elaydi, Rafael Luís, Henrique Oliveira
Mathematics Faculty Research
This survey contains the most updated results on the dynamics of periodic difference equations or discrete dynamical systems this time. Our focus will be on stability theory, bifurcation theory, and on the effect of periodic forcing on the welfare of the population (attenuance versus resonance). Moreover, the survey alludes to two more types of dynamical systems, namely, almost periodic difference equations and stochastic di®erence equations.