Mathematics Colloquium SeriesCopyright (c) 2022 Nova Southeastern University All rights reserved.
https://nsuworks.nova.edu/mathematics_colloquium
Recent documents in Mathematics Colloquium Seriesen-usWed, 26 Oct 2022 02:37:25 PDT3600One iteration for the second boundary condition for the nonlinear one dimensional Monge-Ampere equation
https://nsuworks.nova.edu/mathematics_colloquium/ay_2022-2023/events/2
https://nsuworks.nova.edu/mathematics_colloquium/ay_2022-2023/events/2Thu, 03 Nov 2022 12:30:00 PDT
The design of lenses and mirrors, in free form i.e. with no a priori symmetry assumption, has a long list of applications including materials processing, energy concentrators, medicine, antennas, computing lithography, laser weapons, optical data storage, imaging etc. The design process can be reduced to solving a generalized Monge-Ampere equation where the unknown is a function with a convexity property and subject to a constraint that a generalized gradient maps a given domain onto a prescribed one. The latter type of constraint is known as second boundary condition. The model one dimensional Monge-Ampere equation is nonlinear in the first order derivatives. We show that, although the discrete problem is nonlinear in the first order derivative, it can be solved with just one iteration. We also illustrate how the second boundary condition is reformulated in terms of the asymptotic cone of a convex extension.
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Gerard AwanouOn the Linear Independence of Finite Gabor and Wavelet Systems
https://nsuworks.nova.edu/mathematics_colloquium/ay_2022-2023/events/1
https://nsuworks.nova.edu/mathematics_colloquium/ay_2022-2023/events/1Thu, 15 Sep 2022 12:30:00 PDT
Gabor and Wavelet Systems are some of the most important families of integrable functions with great potential in applications. Those applications include numerical analysis, signal processing (sound, images), and many other areas of physics and engineering. In this talk, we will present some partial results on a conjecture that states each finite Gabor system is linearly independent. We will also present cases of linearly independent and cases of linearly dependent finite wavelet systems.
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Abdelkrim BourouihiyaA Novel TCR Clustering Method for SARS-COV-2 Epitopes
https://nsuworks.nova.edu/mathematics_colloquium/ay_2021-2022/events/6
https://nsuworks.nova.edu/mathematics_colloquium/ay_2021-2022/events/6Fri, 22 Apr 2022 12:00:00 PDT
T-cell epitopes are peptides generated from antigens that are presented by MHC class I and class II molecules to T-cells. These epitopes are usually identified by T-cell receptors (TCRs) of CD4 T-cells which then causes transformation of CD4 T-cells to helper or regulatory T cells. Recently, there has been growing interest in the role of T cells and their involvement in various ailments including SARS-COV-2, cancer, autoimmune diseases and other infectious diseases. However, the mechanism of TCR epitope recognition by Tcell receptors (TCRs) of CD4 T-cells at a repertoire level is still not fully understood. In this project, we reviewed standard TCR clustering methods and developed a novel TCR clustering technique for two SARSCOV- 2 epitopes. Using Principal Component Analysis, we analyzed twenty different physiochemical properties of amino acids and then converted the amino acids in the TCR sequences to numerical strings. We then used four distances methods (Cosine, Cityblock, Euclidean and Correlation) on these strings, and clustered the TCRs in order to compare the dendrogram outputs and see which method does a better job of grouping together like TCRs activated by each epitope. Results were compared to standard matrices such as BLOSUM, PAM, and Gonnet. We thus present a novel TCR clustering technique that will be less computationally strenuous and more cost-effective compared to traditional methods and can be easily utilized by the scientific community to learn more about TCR repertoire sequencing.
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Naziba A. NuhaA Weighted Probability Measure for Objects in Euclidean Space
https://nsuworks.nova.edu/mathematics_colloquium/ay_2021-2022/events/5
https://nsuworks.nova.edu/mathematics_colloquium/ay_2021-2022/events/5Thu, 21 Apr 2022 12:30:00 PDT
Since we were little kids, we developed our own sense dimension as a measure of some kind of extent. Whether it be length, width, or height, we intuitively understand how these features fit in the three-dimensional world we live in, and how to measure it. Nevertheless, mathematicians have found themselves dealing with objects, like fractals, and spaces, like R4 , that challenge our intuitive and self-developed definition of measure, to the point that it is not sufficient anymore. Lebesgue measure and Harsdorf measure for example are ways of assigning a measure to objects that belong to n-dimensional Euclidean spaces, in an effort to find better and more general methods to compare figures, quantities, and masses. Here, I have compared spaces of concentric circles (targets), and studied their behavior in terms of probability masses, varying the number of targets, the size of the circles, and the dimension they belong (plane, space, R4 , Rn).
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Alessandro XelloModeling and simulation of microscopic fibers in a viscous fluid
https://nsuworks.nova.edu/mathematics_colloquium/ay_2021-2022/events/4
https://nsuworks.nova.edu/mathematics_colloquium/ay_2021-2022/events/4Thu, 14 Apr 2022 12:30:00 PDT
In biology, the movements of tiny structures often rely on the mechanical properties of long, thin tubes. For example, bacteria swim by rotating their flagella, and in cell division (mitosis) the two copies of the DNA must be pulled apart by microtubules. To understand these processes it is very tempting to take advantage of the large aspect ratio of the thin structures, for example by modeling them as one-dimensional curves rather than as more complicated objects with volume and surface area. This kind of shortcut saves a lot of work! I will describe one standard and widely used tool known as "Slender Body Theory" and then share some new modeling and simulation results illuminating the circumstances under which SBT is or is not an accurate approximation.
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William MitchellHow Prey Defense Patterns Predator-Prey Distributions
https://nsuworks.nova.edu/mathematics_colloquium/ay_2021-2022/events/3
https://nsuworks.nova.edu/mathematics_colloquium/ay_2021-2022/events/3Thu, 24 Mar 2022 12:30:00 PDT
In ecology, predator and prey species share a common interest in survival. However, this common interest places these species at odds with each other. Predators need to consume prey for their survival. Prey, on the other hand, do not survive if they are consumed. To meet their needs, predators engage in foraging or prey-taxis behaviors whereby they seek areas of high prey density. For prey there are numerous defense strategies to engage including aposematic mechanisms to advertise they are not worth the predator’s while, attacking the predator through chemical or community defense mechanisms, and alarm calls to seek assistance from predators at` higher trophic levels of the food chain; to name a few. In this talk, we will focus on competition between prey-taxis and chemical defense; placing a particular emphasis on conditions leading to spatial segregation between predator and prey, or as it is known in mathematics, pattern formation. We will also discuss other prey defense mechanisms such as the burglar alarm hypothesis and the potential impact of prey defense mechanisms on prey species in resource competition.
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Evan HaskellNumerical schemes for integro-differential equations related to alpha-stable processes
https://nsuworks.nova.edu/mathematics_colloquium/ay_2021-2022/events/2
https://nsuworks.nova.edu/mathematics_colloquium/ay_2021-2022/events/2Thu, 10 Mar 2022 12:30:00 PST
The mean first exit time, escape probability and transitional probability densities are utilized to quantify dynamical behaviors of stochastic differential equations with non-Gaussian, α-stable type Lévy motions. Taking advantage of the Toeplitz matrix structure of the time-space discretization, a fast and accurate numerical algorithm is proposed to simulate the nonlocal Fokker-Planck equations on either a bounded or infinite domain. Under a specified condition, the scheme is shown to satisfy a discrete maximum principle and to be convergent. The numerical results for two prototypical stochastic systems, the Ornstein-Uhlenbeck system and the double-well system are shown.
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Xiaofan LiMathematical Modeling of Lung Cancer Screening Studies
https://nsuworks.nova.edu/mathematics_colloquium/ay_2018-2019/events/5
https://nsuworks.nova.edu/mathematics_colloquium/ay_2018-2019/events/5Tue, 12 Feb 2019 12:05:00 PST
Lung cancer has the second highest cancer incidence, second only to prostate cancer in men and breast cancer in women. Furthermore, more cancer deaths are attributable to lung cancer than any other cancer for both genders. There is a high public health need for effective secondary prevention in the form of early detection and early treatment, complementary to smoking cessation efforts. The U.S. National Lung Screening Trial (NLST) demonstrated that non-small cell lung cancer (NSCLC) mortality can be reduced by 20% through a program of annual CT screening in high-risk individuals. However, CT screening regimens and adherence vary, potentially impacting the lung cancer mortality benefit. The mortality benefit attributable to a program of CT screening is largely determined by the natural history of lung cancer progression. Tumor doubling times and the maximum tumor size at which a NSCLC would be curable by early detection (cure threshold) are key factors. In this talk, I will address novel statistical methodology used to estimate parameters governing a stochastic model of the natural history of lung cancer. We estimate the median tumor size at cure threshold among the most aggressive NSCLCs to be between 10-15 mm. We demonstrate consistency of our model with tumor size and stage data from distinct lung screening trials, namely, the Mayo Lung Project, the Mayo CT study and the NLST, in addition to data from SEER, the national cancer registry and highlight key differences between men and women. The majority of NSCLCs in the NLST were treated at tumor sizes greater than our median cure threshold estimates, consistent with the modest 20% mortality reduction attributable to CT screening observed in the NLST. These results highlight the strong need for novel classification technology that can better distinguish and treat the most aggressive NSCLCs when they are small (i.e. 5-15mm). This talk will also discuss the model’s consistency with the recently announced European NELSON lung screening trial results (October 2018). I will also discuss how novel DNA sequencing technologies may be incorporated into a lung screening regimen in order to improve outcomes.
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Deborah L. GoldwasserFrom Derivation to Error Analysis of Splitting Methods—A Contemporary Review
https://nsuworks.nova.edu/mathematics_colloquium/ay_2021-2022/events/1
https://nsuworks.nova.edu/mathematics_colloquium/ay_2021-2022/events/1Thu, 10 Feb 2022 12:30:00 PST
Splitting methods, with representative examples such as ADI (alternating-direction implicit) method and LOD (local one-dimensional) method, have been playing a significant role for the numerical solution of differential equations. In this talk, we will start from a seemed-to-be obvious issue as an introduction of the modern splitting methods. Historical roots of the literature will be mentioned. We will then use a splitting approach for solving a semi-linear Kawarada partial differential equation which is extremely important to numerical combustion, environmental protection, and biomedical research. Finally, the concept of global error and its estimates will be discussed and extended.
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Qin ShengWorld Statistics Day: Malaria and its Effects on the World: A Statistical Look
https://nsuworks.nova.edu/mathematics_colloquium/ay_2020-2021/events/3
https://nsuworks.nova.edu/mathematics_colloquium/ay_2020-2021/events/3Tue, 20 Oct 2020 12:30:00 PDT
As October 20, 2020 is designated United Nations World Statistics Day, we look at an important statistical problem using a data set collected by researchers from the United Nations. We have all heard about Malaria and seen the effects it could have on friends and family. Still, while we ponder on the who and why this could have occurred, we are here to tell you about the what and how. The severity of this disease can be seen throughout the world. In this presentation, we will look at the number of reported cases of Malaria worldwide and how they affected these different countries through time, in addition to the preexisting geopolitical and economical status of each.
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Aysha Nuhuman et al.Quaternions and Matrices of Quaternions
https://nsuworks.nova.edu/mathematics_colloquium/ay_2020-2021/events/2
https://nsuworks.nova.edu/mathematics_colloquium/ay_2020-2021/events/2Wed, 28 Oct 2020 12:00:00 PDT
Quaternions comprise a noncommutative division algebra (skew field). As part of contemporary mathematics, they find uses not only in theoretical and applied mathematics but also in computer graphics, control theory, signal processing, physics, and mechanics. Speaker, N S U Professor, Fuzhen Zhang reviews basic theory on quaternions and matrices of quaternions, presents important results, proposes open questions, and surveys recent developments in the area.
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Fuzhen ZhangClassic/Quantum Harmonic Oscillator
https://nsuworks.nova.edu/mathematics_colloquium/ay_2020-2021/events/1
https://nsuworks.nova.edu/mathematics_colloquium/ay_2020-2021/events/1Wed, 18 Nov 2020 12:00:00 PST
A Harmonic Oscillator is an integral part of periodic motion in Classical and Quantum Theory. For systems with small fluctuations near stable points of equilibrium, the Harmonic Oscillator serves as a good approximation for measuring eigenstates and wave amplitudes of the particle(s). Aside from the classical version, this presentation will include the Lie Algebra of commutation relations as well as the ladder operators (Discrete and Continuous) as it pertains to a Quantum Harmonic Oscillator. After that, one of its' contributions to scalar fields in Quantum Field Theory, namely the Casimir Force, will be discussed. Whether it is a system of one oscillator or a system of decoupled oscillators, this concept could be applied to the fields of Quantum Field Theory and Mathematical Physics.
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Killian J. HitsmanTensor Eigenvalue Problems and Modern Medical Imaging
https://nsuworks.nova.edu/mathematics_colloquium/ay_2019-2020/events/8
https://nsuworks.nova.edu/mathematics_colloquium/ay_2019-2020/events/8Thu, 20 Feb 2020 12:35:00 PST
Tensors (or hypermatrices) are multidimensional generalization of matrices. Although historically they are studied from the perspective of combinatorics and (hyper)graph theory, recent progress in the subject shows how useful they are in more applied sciences such as physics and medicine. In this presentation, I introduce a few tensor eigenvalue problems and their application to higher order diffusion tensor imaging such as diffusion-weighted magnetic resonance imaging (DW-MRI) and higher angular resolution diffusion imaging (HARDI).
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Vehbi Emrah PaksoyUsing Slow-Fast Dynamical Systems to Understand Regime Shifts in Ecology
https://nsuworks.nova.edu/mathematics_colloquium/ay_2019-2020/events/7
https://nsuworks.nova.edu/mathematics_colloquium/ay_2019-2020/events/7Fri, 04 Oct 2019 12:05:00 PDT
In ecology, regime shifts are continual rapid change between different long-lasting dynamics. For instance, rapid evolutionary changes have been observed in a wide variety of organisms, both in predators and in prey. Another example is disease outbreak, where a system exhibits qualitative changes after long periods of apparent quiescence. Using the theory of slow-fast dynamics, for systems of differential equations with sufficiently large separation of time scales we derive conditions under which regime shifts occur. This is joint work with Shigui Ruan and Gail Wolkowicz.
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Ting-Hao HsuAlgebraic Frames and Ultrafilters
https://nsuworks.nova.edu/mathematics_colloquium/ay_2019-2020/events/6
https://nsuworks.nova.edu/mathematics_colloquium/ay_2019-2020/events/6Fri, 01 Nov 2019 12:05:00 PDT
A frame, also known as pointfree topology, is a complete lattice that satisfies a strong distributive property, known as the 'frame law.' Originally, the study of frames began as studying topological spaces without points, hence the name pointfree topology. Due to this connection, different topological concepts can be generalized to frames, for example, compactness. In the first part of the talk, I will explain the basic notions of frames and their connection with topology. It turns out that we can find frame structure in other categories than topological spaces. For example, given a commutative ring R with identity, the lattice of radical ideals of R, Rad(R), is a frame. As a result, concepts from ring structure can also be generalized to frames, for example, primes and minimal primes, annihilators, etc. I will discuss some of these concepts in the language of frame theory. In the last part of the talk, I will describe filters and ultrafilters on frames and show their connections with certain prime structures of a frame.
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Papiya BhattacharjeeHow Mathematics Can Help Winning the War Against Cancer?
https://nsuworks.nova.edu/mathematics_colloquium/ay_2019-2020/events/5
https://nsuworks.nova.edu/mathematics_colloquium/ay_2019-2020/events/5Fri, 18 Oct 2019 12:05:00 PDT
In this talk, I will present a few mathematical models that aims to understand how our immune system interact with cancer cells. In particular, we focus on a model that studies the role or regulatory T cells. Recent advance in the field of regulatory T cell reveals that it plays a vital role during immunotherapy. For example, a higher ratio between regulatory T cells and effector T cells within tumor tissue is associated with worse prognoses in many cancers, including ovarian cancer (Leffers et al., 2009), lung cancer (Tao et al., 2012), glioblastoma (Sayour et al., 2015). On the other hand, the tug war between regulatory T cells and effector T cells for interleukin-2 may chisel immune response against cancer. In this talk, we demonstrate mathematically, for the first time, that the initial ratio between regulatory T cells and effector T cells does impact the tumor recurrence time. We also demonstrate the effectiveness of utilization of IL-2 may flip the outcome of immunotherapy, providing further evidence that it may be clinically viable to modulate the consumption of IL-2 by Tregs.
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Peng FengSoccer Tournament Matrices
https://nsuworks.nova.edu/mathematics_colloquium/ay_2019-2020/events/4
https://nsuworks.nova.edu/mathematics_colloquium/ay_2019-2020/events/4Thu, 30 Jan 2020 12:35:00 PST
In this talk, I will present a combinatorial object, soccer tournament matrices, which is understandable to undergraduate students and gives a taste of combinatorial matrix theory. Consider a round-robin tournament of n teams in which each team plays every other team exactly once and where ties are allowed. A team scores 3 points for a win, 1 point for a tie, and 0 point for a loss, then each particular result leads to a soccer tournament matrix. Let T(R, 3) denote the class of all soccer tournament matrices with the row sum vector R. In this talk, I will explore some necessary conditions of a vector R, such that T(R, 3) is nonempty with the audience, and then for some R, I will show an algorithm to construct a soccer tournament matrix whose row sum is R.
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Lei CaoMinimal Rank Completions of Partial Matrices
https://nsuworks.nova.edu/mathematics_colloquium/ay_2019-2020/events/3
https://nsuworks.nova.edu/mathematics_colloquium/ay_2019-2020/events/3Fri, 15 Nov 2019 12:00:00 PST
Completion problems for partial matrices are defined and partial matrices are associated to bipartite graphs. Minimal ranks for scalar and block partial matrices with simple structures are presented. Calculating the minimal rank is classified as an NP-hard problem, what means that in general it is very difficult to calculate the minimal rank of a unstructured block (scalar) partial matrix. A conjecture states that the minimal rank of a partial matrix has an exact formula if and only if the associated bipartite graph is chordal. We present some upper estimates for the case that the associated bipartite graph is a single cycle, the most simple non-chordal case. The symmetric cyclic case is also treated.
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Edgar PereiraHow Good Are Standard Copulas Anyway?
https://nsuworks.nova.edu/mathematics_colloquium/ay_2019-2020/events/2
https://nsuworks.nova.edu/mathematics_colloquium/ay_2019-2020/events/2Thu, 07 Nov 2019 12:25:00 PST
First, we will raise a question: How good are standard copulas in capturing the dependency structure? To this end we will offer a series of simulated/numerical examples demonstrating that, more often than not, standard model copulas do not capture the underlying dependency structure. We believe that copula models, unlike other statistical tools, are too readily accepted by practitioners. Rigorous, goodness-of-fit tests are commonly replaced by off-hand statements like: “it works well”. To this end, the second part of the talk offers a theoretical result, an umbrella type theorem tailored for creating numerous Goodness of Fit tests for copulas.
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Dragan RadulovicEigenvalue inequalities of matrix product
https://nsuworks.nova.edu/mathematics_colloquium/ay_2019-2020/events/1
https://nsuworks.nova.edu/mathematics_colloquium/ay_2019-2020/events/1Fri, 20 Sep 2019 12:05:00 PDT
Given two n-by-n complex matrices, one is Hermitian and one is positive semidefinite, all of the n eigenvalues (counting multiplicities) of the product of the given matrices are necessarily real. Selecting any k of the n eigenvalues, we present lower and upper bounds for the sum of these k selected eigenvalues. Our results extend and complement the existing ones.
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Fuzhen Zhang