# Mathematics.

Modeling, Analysis and Simulations of Debonding of a Bonded Rod-Beam System Caused by Mechanical Stresses, Humidity and Thermal Effects. Pawel Marcinek and Meir Shillor, Oakland UniversityThere is a considerable industrial interest in developing lightweight materials, such as metals with low density (magnesium, aluminum) or polymer materials in new parts and components to make them stronger and more fuel economic. Lightweight materials require a special bonding technology, especially for joining dissimilar materials which cannot be bolted or welded. It is observed that the bonding strength decreases in time due to mechanical stresses, thermal and humidity effects. Therefore, there is a need for mathematical models to gain deeper understanding of the deterioration process of adhesives and to qualitatively predict it. In this lecture, we will present the model for the process of debonding of two slabs that are glued together. We will discuss the physical and mathematical details of the model, especially we will state the theorem of existence and uniqueness of the weak solution. We will also analyze the details of the numerical algorithm used to find approximate solutions for the model and its stability. Finally, we will present extensive numerical simulations of the model behavior and the dynamics of the debonding process.

Involute and Evolute of Rectifying Curves in [R.sup.3]. Yun Myung Oh, Andrews University

The idea of rectifying curves in [R.sup.3] was introduced by B. Y. Chen in 2003 and many characterizations results have been found with applications in kinematics and mechanics. The idea has been generalized to the arbitrary dimension and it is now called the rectifying submanifold. In this project, we are going to investigate the involute and evolute of rectifying curves in [R.sup.3] and find some characterizations of the curve.

Perturbation of a Nonlinear Elliptic Mathematical Model. Joon Hyuk Kang, Andrews University

Previous work established sufficient conditions for the uniqueness of the positive solution to the general elliptic system. That is, we proved that under certain conditions, the system has the unique positive solution at fixed rates functions.

In this research, we extended the uniqueness results for the modified system obtained by perturbing rates functions using Implicit Function Theorem, Maximum Principles, and Spectrum Estimates.

Adapting Adaptive Remedial Mathematics. Lynelle Weldon, Andrews University

The remedial mathematics courses at Andrews University use individualized curriculum with the ALEKS program as the stand-alone content resource augmented by instructor and peer lab assistant tutoring. This system has various structural components such as weekly hours requirements, all work in a notebook, and coaching in goal-setting and tracking. One persistent challenge (for a larger-than-desired proportion of students) is student engagement. This last semester I piloted an adaptation of this course with some whole-class learning activities. I will report on the results and future plans.

A Mathematical Model for Chagas Disease Dynamics in the Gran Chaco with Sylvatic Transmission and Vector Life Stages. Anna Maria Spagnuolo and Kathryn Anderson, Oakland University; Ryan Capouellez, University of Michigan; Daniel J. Coffield Jr., University of Michigan-Flint; Meir Shillor, Oakland University; Gabrielle Stryker, Central Washington University

Chagas disease is a parasitic vector-borne illness which infects mammals, including humans, and exists predominantly in Latin and South America. This paper will present a mathematical model consisting of 29 coupled differential equations, some with delays, which attempts to characterize the key aspects of Chagas disease dynamics in the Gran Chaco region of South America. For an example village, these equations model the population of vectors in the domestic and peridomestic regions, infected vectors in the domestic and peridomestic, as well as susceptible and infected humans, infected dogs, and infected mammals. As an addition to this model, an equation describing wild populations of vectors (sylvatic) and transfer to the domicile from these populations is included. This model also attempts to create a more accurate portrayal of the vector populations by including the presence of vector nymph stages into all vector populations (except the sylvatic). The main interest for this work is to provide a tool in the form of computational simulations to test different scenarios that will aid researchers in potentially discovering and exploring avenues that will reduce disease incidence in humans.

Cannibalism and Synchrony in a Periodic Matrix Seabird Population Model. Mykhaylo M. Malakhov, Andrews University; Benjamin MacDonald, University of Vermont; Shandelle M. Henson, Andrews University; J. M. Cushing, University of Arizona

Increases in sea surface temperatures (SSTs) in the Pacific Northwest lead to food resource reductions for seabirds and have been correlated with several marked behavioral changes. Particularly, higher SSTs lead to increased egg cannibalism among gulls, which in turn promotes every-other-day egg-laying synchrony in the colony. We pose and analyze a discrete-time periodic matrix population model to study the effect of these changes on the long-term survival and dynamics of the gull population. We show that cannibalism and synchrony can lead to backward bifurcations and strong Allee effects, which allow the population to survive at lower resource levels than would be possible otherwise.

Bifurcations in a Discrete-Time Model for Synchronous Egg-Laying in a Seabird Colony. Christiane Gallos, Dorothea Gallos, Whitney Watson, and Shandelle M. Henson, Andrews University

Glaucous-winged gulls (Larus glaucescens) that breed in a large colony on Protection Island, Washington are known to exhibit every-other-day egg-laying synchrony in dense areas of the colony. In this talk, we present a discrete-time model of egg-laying behavior. We then find the steady states of the model, find the conditions under which these steady states are stable, and determine whether, in this model, egg-laying synchrony is beneficial at the population level.

Milnor Invariants Vanish for Shake Slice Links. Anthony Bosnian, Andrews University

A link is an embedding of circles into 3-dimensional space, possibly with components knotted or linked. The Milnor invariants are a well-studied measure of the linking and higher order linking between the components of a link. These invariants all vanish for slice links, an important family of links in low dimensional-topology. A generalization of slice links, called shake slice links, relate to the problem of finding embeddings of spheres in 4-dimensional manifolds. We show that the Milnor invariants vanish for shake slice links.

Effects of Using Online Applets on Students' Learning of Trigonometric Functions. Mustafa Demir, University of Detroit Mercy

This study examined the impact of using web-based applets on students' knowledge of trigonometric functions. Having a solid understanding of trigonometric functions is crucial to learn various advanced mathematical concepts; such as identifying geometric properties of complex numbers and using sinusoidal functions to model many periodic phenomena in science and engineering. Students often have difficulties in making connections between right triangle and unit circle approaches (Martinez-Planell and Delgado, 2016), and conceptualizing a circle's radius as a unit of measure (Moore, LaForest, and Kim, 2012). Numerous studies already examined the integration of online applets into mathematics instruction, however few studies analyzed the effects of using online applets on students' knowledge of trigonometry. Thus, the study examined college students' learning of trigonometric functions after using a set of online applets in their Pre-Calculus course. In particular, sixty-two college students' (Experimental: 30 and Control: 32) final exam papers were analyzed. The findings revealed that students using online applets in their courses showed higher performance in identifying geometric properties of trigonometric functions than their peers in the control group. In contrast, control group students ohtained higher scores on the questions requiring the use of trigonometric identities than the students in the experimental group.

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Title Annotation: | bonded rod-beam analysis, rectifying curves in Euclidean space, perturbation of a nonlinear elliptic mathematical model |
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Publication: | Michigan Academician |

Date: | Sep 22, 2018 |

Words: | 1226 |

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