The Michigan Section of the Mathematical Association of America and MichMATYC 92nd Annual Meeting
Hillsdale College Hillsdale, Michigan April 1–2, 2016 Michigan Section of the Mathematical Association of America
2015-2016 Officers and Staff Chair Brian Snyder, Lake Superior State University 4-Yr Vice Chair Gavin LaRose, University of Michigan, Ann Arbor 2-Yr Vice Chair Jan Roy, Montcalm Community College Sec/Treas Mark Bollman, Albion College Governor Matt Boelkins, Grand Valley State University Past Chair Michele Intermont, Kalamazoo College Newsletter Ed Victor Piercey, Ferris State University Co-Dir. MMPC Kim Rescoria, Eastern Michigan University Carla Tayeh, Eastern Michigan University Webmaster Paul Pearson, Hope College Liaison Coord David Austin, Grand Valley State University PIO Robert Xeras, Siena Heights University, retired
2016 Annual Meeting Program Committee Chair Gavin LaRose, University of Michigan, Ann Arbor Members Dave Gaebler, Hillsdale College Jan Roy, Montcalm Community College
2016 Annual Meeting Local Arrangements Committee Chair Dave Gaebler, Hillsdale College Members David Murphy, Hillsdale College
MichMATYC 2015-2016 Officers and Staff President Cindie Wade, St. Clair Community College Past President Jack Rotman, Lansing Community College Sec/Treas Jeff Morford, Henry Ford Community College Newsletter Ed Khadija Ahmed, Monroe County Community College Affiliate Delegate Jan Roy, Montcalm Community College Affiliate Delegate Marie St. James, St. Clair Community College 2016 Annual Meeting of the Michigan Section of the MAA & MichMATYC The 2016 Annual Meeting of the Michigan Section of the Mathematical As- sociation of America and the Michigan Mathematical Association of Two- Year Colleges is being held on Friday and Saturday, April 1 and 2, on the campus of Hillsdale College in Hillsdale, Michigan. The registration desk and exhibits are in the Searle Center, with registration beginning at 1:15pm on Friday. Talks are in the Dow Center and Lane Hall. A map of campus is available in the registration materials provided with this program, or on-line at
Program Notes: Pages 3–5 give the timetable for this year’s program, and pages 6–17 provide abstracts. Talks by undergraduates are denoted with an asterisk “∗,” and the talks by graduate students are denoted with a double dagger “‡.”
Meals: (Advance registration is required. Contact Dave Gaebler with ques- tions:
TeMaCC Workshop: TeMaCC (Teaching Mathematics Content Courses), a special interest group of the MI-AMTE, is organizing a special working group for this conference. The working group will focus on the following questions:
1 • What key pillars of geometry courses for future K-8 teachers might be enhanced by the use of GeoGebra? • What mathematical, pedagogical, and cognitive considerations are there in selecting or designing GeoGebra applets for use in these courses? The group will analyze existing GeoGebra applets and collaboratively start work on their own. Both experienced and novice GeoGebra users are wel- come and encouraged to participate. They hope to foster a discussion and collaboration that will continue beyond the end of the conference. Pre- registration is needed; Section meeting attendees who are interested in the workshop should contact Nina White at
Lodging: The following hotels are recommended: ◦ The campus hotel (the Dow Center; on campus, (517) 437-3311). Rooms are $100–$150. ◦ Days Inn (3241 W Carleton Rd.; 517-439-3297). Rooms are $80, one queen bed, double occupancy.
Student Activities: The Ron Mosier Memorial Award will be presented at the closing awards session to the student(s) with the most outstanding talk. Saturday there will be a pizza lunch for the MUMC.
Internet Access and Computers: Wireless networking is available on the Hillsdale campus. The wireless network is called CampusNet and the password is liberty1844. In addition, there is a computer lab in the basement of Lane Hall, right next to the talks. The username and password to log in to these computers are both maaguest. This is also the account for the classroom computers attached to projectors.
Sponsors: McGraw Hill and Cengage have generously sponsored our coffee breaks.
2 2016 Joint MAA/MichMATYC Meeting, Hillsdale College
Friday, April 1, 2016 1:15-2:15 Registration (Searle Center) Welcome and Opening Plenary (Dow Center A & B) 2:15-3:20 Marty Golubitsky, Ohio State University/MBI Patterns of Synchrony: From Animal Gaits to Binocular Rivalry
Local Invited Lecture Local Invited Lecture (Lane 124) (Lane 125) 3:30-4:10 Christine Phelps, Central Michigan U David Murphy, Hillsdale College Improving Mathematics Teaching by Tinkering Desingularizations of Some Nilpotent Orbit Closures Room Lane 124 Lane 125 Lane 123 Brian Chadwick (Michigan State U) Joon Kang (Andrews U) Implementing a Summer Bridge Program at 4:15-4:35 A General Elliptic Nonlinear System of Two Michigan State University to Prepare Students Functions with Application for College Algebra
Yun Oh (Andrews U) TeMACC Workshop On the Riemannian Submersion Invariant and Clark Wells (Grand Valley State U) 4:40-5:00 Lagrangian Submanifolds The Case for Proofs that Explain
5:10-5:30 Business Meeting (5:10-5:40) 5:30-6:30 Social Hour (Searle Center) Banquet and Plenary Lecture (Searle Center) 6:30-9:00 Alissa Crans, Loyola Marymount University Musical Actions of Dihedral Groups Saturday, April 2, 2016 8:00-8:40 Light Breakfast (Dow Center A & B) and Registration (Searle Center) Plenary Lecture (Dow Center A & B) 8:45-9:40 Hortensia Soto, University of Northern Colorado Developing Students' Mathematical Reasoning: Using Notions of Embodied Cognition
Local Invited Lecture Local Invited Lecture (Lane 125) (Lane 124) 9:50-10:25 Andrew Ross, Eastern Michigan U Cam McLeman, UM Flint Matching EMU's Quantitative Reasoning Course with the An Entirely Random Approach to Algebraic Number Theory Michigan Transfer Agreement
10:30-11:10 Break (Lane Corridor) Michigan Undergraduate Math Contributed Papers Conference Room Lane 233 Lane 125 Lane 124
Arundhati Misra (Saginaw Valley State U) and Hyeona * Mohit Bansil, Aaron Craig and Lim (Mississippi State U) Paul Pearson (Hope C) Nicholas Paul (LTU) 11:10-11:30 Nonlocal Speckle Denoising Linear Algebra Ought To Be Math in Your Bath: An TeMACC Workshop Model Based on Nonlocal In Pictures Application of the Heat Equation (Lane 123) Means of Similar Neighborhoods
Feryal Alayont and David Alexander Israetel (Davenport * Ethan Bush (UM Flint) Clark (Grand Valley State U) U) An Analogue of the Median Service Learning for 11:35-11:55 A Trigonometric Model of Voter Theorem in Approval Pre-service Teachers in Continuous Probability Voting Mathematics Content Distributions Courses
MUMC Pizza Lunch Conference Luncheon 12:00-12:45 (Dow Gillespie Room) (Searle Center) Plenary Lecture (Searle Center) 12:55-1:50 Karen Smith, University of Michigan Ann Arbor
Room Lane 233 Lane 125 Lane 124 Victor Piercey and Roxanne Cullen (Ferris * Daniel Slonim (Hillsdale C) ǂ Emily Olson (Michigan State U) State U) 2:00-2:20 Interleaving of Path Sets Properties of the Linear Extension Poset An Unlikely Adventure: Linked Math and English
* Brooke Szymoniak (SVSU) Morgan Fonley (Alma C) ǂ Matthew Plante (Central Michigan U) 2:25-2:45 On the Existence of Normal Invisible H20: Tracking the Water We Counting Anosov Graphs Subgroups of Prime Index Cannot See
* Sarah Petersen (Hope C) Level Curves of a Real Algebraic ǂ Pin-Hung Kao (Central Michigan U) Barbara Britton (Eastern Michigan U) 2:50-3:10 Function: A Generalization of a On Polynomials at Prime Arguments The Pythagorean Comma: Pause for Music Theorem of Pólya
* Anthony Pecoraro (GVSU) Curtis Grosse and Juan Sancen (SVSU) Classifying 7 Dimensional A Model for Superior Risk-Adjusted 3:15-3:35 Indecomposible Solvable Lie Returns Using Statistics and Available Algebras with Niradical Financial Tools Isomorphic to A5,2⊕ R 3:35-4:00 Break (Lane Corridor) Closing and Student Awards (Dow Center A & B) 4:00-5:15 Plenary Panel Math Circles and Math Teacher Circles in Michigan 2016 Annual Meeting of the Michigan MAA and MichMATYC Speaker and Abstract List
INVITED PLENARY LECTURES
Martin Golubitsky, Ohio State University/MBI Opening Plenary Dow Center A & B Patterns of Synchrony: From Animal Gaits to Binocular Rivalry This talk will review previous work on quadrupedal gaits and recent work on a generalized model for binocular rivalry proposed by Hugh Wilson. Both applications show how rigid phase-shift synchrony in periodic solutions of coupled systems of differential equations can help understand high level collective behavior in the nervous system.
Alissa S. Crans, Loyola Marymount University Friday Banquet Searle Center Musical Actions of Dihedral Groups Can we hear an action of a group? Or a centralizer? In the same way it is possible to see group structure in a crystal, it is also possible to hear group structure in music. We will investigate two ways that the dihedral group of order 24 acts on the set of major and minor chords and illustrate both geometrically and algebraically how these two actions are dual. Both actions and their duality have been used to analyze works of music as diverse as that of Beethoven and the Beatles. (This is joint work with Thomas M. Fiore and Ramon Satyendra.)
Hortensia Soto, University of Northern Colorado Saturday Morning Plenary Dow Center A & B Developing Students’ Mathematical Reasoning: Using Notions of Embodied Cognition The learning theory of embodied cognition asserts that thought is based in perception and action. In this talk, I will describe embodied cognition and its role in developing abstract mathematical notions. In order to better understand the underpinnings of this perspective, we will conduct a little experiment. My hope is that this presentation will offer another avenue for instructors to explore and to develop their teaching.
6 Karen Smith, University of Michigan Ann Arbor Saturday Lunch Plenary Searle Center Noether’s Legacy: Rings and Geometry I will discuss how Noether’s important idea of a ring homomorphism un- derlies deep techniques used in algebraic geometry to understand the sin- gularities of complex algebraic varieties. Several ring homomorphisms are involved as we reduce mod p and then iterate the pth power map to prove that certain classes of varieties have nice resolutions of singularities.
7 LOCAL INVITED TALKS
Christine Phelps, Central Michigan University Friday April 1, 3:30–4:10 Lane 124 Improving Mathematics Teaching by Tinkering Mathematics education research results can be difficult to apply to the col- lege mathematics classroom because teaching is highly dependent on con- text and content (Hiebert, Gallimore, & Stigler, 2002). Some researchers have instead argued instructors should treat teaching as an ongoing exper- iment (Hiebert, Morris, & Glass, 2003). In this talk, I present one method for systematically experimenting with and improving teaching. I give an example from my own teaching with prospective elementary teachers and show how, over time, I have experimented and tinkered to improve prospec- tive teachers’ learning. The lesson I present was taught in an introductory mathematics con- tent course for prospective elementary teachers. The goal of the lesson was to help prospective teachers learn to mathematically analyze a tran- script from an elementary classroom. By tinkering with the way I taught this lesson, I have discovered how to improve prospective teachers learning. For example, when I first designed my lesson, I included group work for prospective teachers to discuss their thoughts together. However, through experimentation, I discovered that group work did not “bring out the best” in prospective teachers’ work. Through experimentation and iterative tin- kering, I have slowly learned how to better support prospective teachers in this lesson.
David Murphy, Hillsdale College Friday, April 1, 3:30–4:10 Lane 125 Desingularizations of Some Nilpotent Orbit Closures Let G be a reductive algebraic group defined over an algebraically closed field k of characteristic p, which we assume is good for G. An involution θ of G determines the fixed point subgroup K = Gθ and a decomposition g = k ⊕ p of g = Lie(G) into ±1-eigenspaces for dθ. Here k = Lie(K), while the (−1)-eigenspace p is the “infinitesimal” symmetric space which identifies with the tangent space at the identity to the symmetric space G/K. Let N (g) be the nullcone of g and set N (p) = p ∩ N (g). The adjoint action of G on N (g) yields an action of K on N (p). Drawing parallels with the well-known and rich study of G-orbits on N (g), we consider desingularizations of orbit closures O for O a K or K◦- orbit in N (p), and discuss applications to topics such as the normality of O and cohomological interpretations of rings of functions for these orbits.
8 This is joint work conducted with Terrell L. Hodge of Western Michigan University.
Cam McLeman, University of Michigan Flint Saturday, April 2, 9:50–10:25 Lane 124 An Entirely Random Approach to Algebraic Number Theory The “class group” of a field of numbers is an abelian group which measures the extent to which that field fails to have the “unique factorization into primes” property we all know and love from the regular integers. The study of class groups dates back to Gauss and Dedekind, and ties to many of the most famous unsolved problems in number theory, and yet so little do we know about these objects that we don’t even know if the trivial group occurs as a class group infinitely often. In this talk, we take advantage of our ignorance and share a new class of results that you get by abandoning any hope of understanding what’s going on deterministically.
Andrew Ross, Eastern Michigan University Saturday, April 2, 9:50–10:25 Lane 125 Matching EMU’s Quantitative Reasoning Course with the Michigan Trans- fer Agreement’s Requirements I will describe the Quantitative Reasoning pathway within the MTA (Michi- gan Transfer Agreement), and how EMU’s main general-education course matches up with it. I will also describe our efforts to assess student perfor- mance across dozens of non-coordinated sections of the class.
9 CONTRIBUTED TALKS Joon Kang, Andrews University Friday, April 1, 4:15–4:35 Lane 124 A General Elliptic Nonlinear System of Two Functions with Application We study mathematical conditions to guarantee the existence of positive solutions to a general non-linear second order system of partial differential equations with homogeneous boundary conditions. This result may apply to illustrate biological conditions under which species of animals residing in the same environment can peacefully coexist forever. Brian Chadwick, Michigan State University Friday, April 1, 4:15–4:35 Lane 125 Implementing a Summer Bridge program at Michigan State University to prepare students for College Algebra For the past 3 years, the mathematics department at Michigan State Uni- versity has developed and implemented a hybrid-format summer bridge program for incoming students addressing their mathematics placement. Students who were close to being eligible for admittance into College Al- gebra were approached and invited to take part in this program. Using WebAssign, we were able to fully and effectively design and develop the online portion of the course for students to gain the skills necessary to put themselves in a position to become eligible for College Algebra in their first semester. Using WebAssign as a course management tool, it became available for video-embedding, attaching documents, pulling problem sets from multiple resources, and personalizing a curriculum so that students maximized their learning experience while progressing through the online portion of the program. Come see the results as well as learn how We- bAssign can be an option for meeting similar goals you have. Yun Oh, Andrews University Friday, April 1, 4:40–5:00 Lane 124 On the Riemannian Submersion Invariant and Lagrangian Submanifolds For a Riemannian submersion π : M n → Bb with totally geodesic fibers, b n ˘ P P 2 the submersion invariant Ax = kAei esk was introduced using the i=1 s=b+1 integrability tensor A of the submersion by B.Y. Chen. He has provided an inequality on this invariant if the manifold M admits an isometric immer- sion into a Riemannian manifold M˜ . In this talk, some recent work on this invariant will be discussed if M admits a Lagrangian isometric immersion.
10 Clark Wells, Grand Valley State University Friday, April 1, 4:40–5:00 Lane 125 The Case for Proofs that Explain Whatever else we may say about proofs, they are certainly a form of writing. One of the most basic tenets of good writing is to know and write to your audience. Over the years, I have realized that the way we typically write (and teach writing of) proofs in textbooks and classrooms does not serve the intended audience nor our educational purposes. In this talk, I will discuss the audience and purpose of proof and make the case that many proofs can be improved by turning them on their heads. I will give examples of proofs that fail to serve their audience and show how they can be rewritten so that they do a better job. ∗ Mohit Bansil, Aaron Craig and Nicholas Paul, Lawrence Tech- nological University Saturday, April 2, 11:10–11:30 Lane 233 Math in Your Bath: An Application of the Heat Equation In order to develop the best strategy for maintaining an even tempera- ture throughout a bathtub, we modeled the diffusion of heat in water. We based our model on the heat equation and divided it into two cases: a two-dimensional base case and a three-dimensional general case. The two- dimensional base case was simplified so that we could write an accurate computer simulation. We used the finite-difference method to approximate the heat transfer modeled by the heat equation. For the general case, we used the three-dimensional heat equation with boundary conditions reflect- ing the state of the tub and room. The results obtained from the base case show how heat diffuses rapidly through turbulent water and water mixed with certain additives. Therefore, we find that the most effective strategy to maintain an even water temperature is to create turbulence in the water and use additives that conduct heat well. Arundhati Misra, Saginaw State University and Hyeona Lim, Mississippi State University Saturday, April 2, 11:10–11:30 Lane 125 Speckle Denoising Model Using Nonlocal Similar Neighborhoods Image denoising is among the most fundamental problems in image pro- cessing. A large range of methods covering various fields of mathematics are available for denoising an image. The initial denoising models are de- rived from energy minimization using nonlinear partial differential equa- tions (PDEs). The filtering based models have also been used for quite a long time where the denoising is done by smoothing operators. The most successful method among them was the nonlocal means proposed by
11 Buades, Coll and Morel in 2005. Though the method is very accurate in removing noise, it is very slow and hence quite impractical.In 2008, Gilboa and Osher extended some known PDE and variational techniques in image processing to the nonlocal framework. The motivation behind this was to make any point interact with any other point in the image. Using nonlo- cal PDE operators, they proposed the nonlocal total variation method for Gaussian noise. Based on this, a nonlocal PDE based speckle denoising model has been developed earlier. The model is faster than nonlocal means but still much slower than the total variation based models. In this pa- per, we develop a faster version of the existing nonlocal PDE based speckle denoising model. We improve the existing model by using similar neighbor- hoods described by Mahmoudi and Sapiro in 2005. For faster convergence, we use the Split Bregman scheme to find the solution to this new model. Paul Pearson, Hope College Saturday, April 2, 11:10–11:30 Lane 124 Linear Algebra Ought To Be In Pictures Even though most introductory linear algebra courses focus on arrows (vec- tors) in real n-dimensional space, linear algebra textbooks commonly focus on symbolic and numerical points of view instead of graphical viewpoints. To develop students’ graphical and conceptual understanding of linear alge- bra, this talk will discuss ways to present coordinate vectors, basis, change of basis, linear transformations relative to non-standard bases, and eigen- stuff graphically, and how to put them into a conceptual framework using grids and commutative diagrams. Time permitting, pros, cons, and chal- lenges in presenting the material this way will be discussed. *Ethan Bush, University of Michigan Flint Saturday, April 2, 11:35–11:55 Lane 233 An Analogue of the Median Voter Theorem in Approval Voting The Median Voter Theorem is a well-known result in social choice theory for majority-rule elections. We develop an analogue in the context of approval voting. On a line, we consider voters to have preference sets that are intervals called approval sets and the approval winner is a point on the line that is contained in the most approval sets. We define median voter by considering the left and right end points of each voter’s approval sets. First we consider the case where approval sets are equal length. We show that if the pairwise agreement proportion is at least 3/4, then the median voter interval will contain the approval winner. We also prove that under an alternate geometric condition, the median voter interval will contain the approval winner, and investigate variants of this result. Our results show there is a way to define conditions where the median voter interval will
12 contain the approval winner thus there exists an analogue of the Median Voter Theorem in approval voting. This talk is based on my joint REU project with Kyle Duke at James Madison University and Miles Stevens at Morehouse College.
Alexander Israetel, Davenport University Saturday, April 2, 11:35–11:55 Lane 125 A Trigonometric Model of Continuous Probability Distributions Products of trigonometric functions such as sine and cosine are used to model unimodal continuous probability distributions. The probability den- sity function is given by f(x) = A(sin x)m(cos x)n, where m and n are positive integers. The values of A are calculated by integration for different values of m and n. The position of the distribution maximum is expressed p in terms of the ratio, m/n, as follows: Xmax = arctan( m/n). Parameters of distributions, such as mean, standard deviation, skewness and kurtosis are computed for different values and ratios of m and n. This simple model can be especially effective for describing skewed and strongly skewed unimodal continuous distributions. The model can also be modified for multi-modal situations. Graphs of standard deviation, skewness and kurtosis for various values of the ratios m/n demonstrate interesting and consistent patterns and are discussed.
Feryal Alayont and David Clark, Grand Valley State University Saturday, April 2, 11:35–11:55 Lane 124 Service Learning for Pre-service Teachers in Mathematics Content Courses Mathematical games played at a Family Math Night are a perfect vehicle for engaging pre-service mathematics teachers and other mathematics majors in service learning opportunities. For the last few years, we have partnered with an area middle school to organize Family Math Nights. At Family Math Nights, K-12 students and their families experience a wide variety of mathematical games and activities in a fun and encouraging setting. As a result, these events serve both as outreach opportunities for our department and great experiences for our pre-service mathematics teachers in a school setting. By having students create their own games based on the content of the mathematics course and/or by having students reflect on their expe- rience in the event, students’ participation in this opportunity can become a service learning project. We will describe the logistics of preparing for a Family Math Night and the two different models we use for incorporating this event as a service learning project into our courses. We will also de- scribe the benefits that we have observed for students at the school and for
13 pre-service teachers, as well as some practical advice for those interested in implementing their own Family Math Nights. *Daniel Slonim, Hillsdale College Saturday, April 2, 2:00–2:20 Lane 233 Interleaving of Path Sets We study an interleaving operation on path sets, which are spaces of one- sided infinite symbol sequences corresponding to the one-sided infinite walks beginning at a fixed initial vertex on a given labeled directed graph. Path sets are useful for the study of intersections of fractals, and have also been used to study problems related to neural networks. Interleaving is a kind of multiplication for path sets that has proved useful in computations. We show how to characterize path sets by finite initial blocks. We then study dynamical and algebraic and properties of interleaving and provide exam- ples. Finally, we discuss a method that will undo the process of interleaving. ‡Emily Olson, Michigan State University Saturday, April 2, 2:00–2:20 Lane 125 Properties of the Linear Extension Poset We use all possible linear extensions of a given poset to build its linear extension poset. If the given poset has n elements and no relations, we obtain the weak Bruhat order on Sn, the symmetric group. If the poset has some relations, the linear extension poset changes dramatically. We will discuss properties of the linear extension poset, including connectedness and maximal elements. Victor Piercey and Roxanne Cullen, Ferris State University Saturday, April 2, 2:00–2:20 Lane 124 An Unlikely Adventure: Linked Math and English Over the last three years, we have been linking a section of a quantitative reasoning course with a section of our first year writing course. By doing so, our students have benefited, often in ways we didn’t expect. In addition, our teaching has grown as what we have learned in one another’s classes has shed new light on how our students think. In this talk, we will share our tale and our insights, along with questions that may guide our collaboration in the future. *Brooke Szymoniak, Saginaw Valley State University Saturday, April 2, 2:25–2:45 Lane 233 On the Existence of Normal Subgroups of Prime Index In this article, we characterize finite groups having normal subgroups of given prime index. Precisely, we prove that if p is a prime divisor of a finite group G, then G has no normal subgroup of index p if and only if
14 G = GGp , where Gp is the subgroup of G generated by all elements of the form gp for any g ∈ G and G is the derived subgroup of G. We also extend a characterization of finite groups with no subgroups of index 2 in J.B. Nganous work “How rare are the subgroups of index 2?” (Mathematics Magazine, Vol. 85, No. 3, June 2012, pp. 215-220) to infinite groups. We display an example to show that for an arbitrary prime index p 6= 2 the characterization does not hold.
‡Matthew Plante, Central Michigan University Saturday, April 2, 2:25–2:45 Lane 125 Counting Anosov Graphs In recent work by Dani and Mainkar, a set of finite simple graphs was used to construct nilmanifolds admitting Anosov diffeomorphisms. In this talk, we report new upper and lower bounds found by Mainkar, Plante, and Salisbury on this particular set of graphs, which we call Anosov graphs. The work presented here will use a convenient equivalence relation to construct a family of quotient graphs. By studying this family we improve a lower bound on the number of Anosov graphs in terms of edges and vertices previously given by Dani and Mainkar, and give lower and upper bounds solely in terms of the number of vertices.
Morgan Fonley, Alma College Saturday, April 2, 2:25–2:45 Lane 124 Invisible H20: Tracking the Water We Cannot See When using mathematical hydrologic models to represent the transmission of precipitation to streamflow, the most easily observed values often repre- sent the outputs of said models, while the observed inputs tend to contain significant uncertainty. Because of this, the ability to work with a hydro- logic model in reverse chronological order (the “backwards direction”) of the water cycle is a useful technique which can result in more accurate models. In this talk, I will present a model for water flow through a hills- lope. I will then apply this model under dry conditions, and use observed streamflow time series to infer the pattern of evaporation which has forced the aforementioned streamflow to occur.
*Sarah Petersen, Hope College Saturday, April 2, 2:50–3:10 Lane 233 Level Curves of A Real Algebraic Function: A Generalization of a Theorem of P´olya We consider an old theorem of George P´olya’s which looks at the level curves of certain polynomial functions and the intersection of such curves with lines of positive slope. We extend P´olya’s theorem, relaxing conditions on the
15 polynomial functions and considering intersections with lines of negative slope. The degree two case gives us hyperbolic paraboloids, leading to a brief visual demonstration of solid analytic geometry (the algebraic study of the real vector space R3). ‡Pin-Hung Kao, Central Michigan University Saturday, April 2, 2:50–3:10 Lane 125 On Polynomials at Prime Arguments Let f ∈ Z[x] be an irreducible polynomial over Z with positive leading co- efficient with degree k. We are interested in the number of prime factors of f(p), where p is a prime. By adopting a method of A.J. Irving with an appli- cation of the Diamond-Halberstam-Richert sieve, we show that the number of prime factors of f(p) can be lowered compared to previous records. In particular, we show that any irreducible quadratic f(p) is a P4 infinitely often. Barbara Britton, Eastern Michigan Univeristy Saturday, April 2, 2:50–3:10 Lane 124 The Pythagorean Comma: Pause for Music There’s an interesting tale about the mathematical choices that must be made when choosing how to set up a piano. It also explains why a good Barbershop quartet can “ring” their chords! *Anthony Pecoraro, Grand Valley State University Saturday, April 2, 3:15–3:35 Lane 233 Classifying 7 Dimensional Indecomposible Solvable Lie Algebras With Ni- radical Isomorphic to A5,2 ⊕ R. This project is the fourth in a series that examine seven-dimensional solv- able Lie Algebras with a six-dimensional niradical. Low dimensional solv- able Lie Algebra classification started back in 1963 by Mubarakzyanov. and were completely classified up to dimension six. A general theorem asserts that if g is a solvable Lie Algebra of dimension n, then the dimension of n its maximum nilpotent ideal (called the nilradical) is at least 2 . For the seven-dimensional algebras, the nilradical’s dimension could be 4, 5, 6 or 7. The four and seven dimensional nilradical cases were classified. We examine the six-dimensional niradical case. We first looked for the six-dimensional nilpotent algebras and found 32 algebras. The first case was completed in 2014, and the second case was completed in 2015. In this project we focus on the class where the nilradical is isomorphic to a direct sum of the five-dimensional algebra A5,1 and the one dimensional algebra denoted by A5,2 ⊕ R.
16 Curtis Grosse and Juan Sancen, Saginaw Valley State University Saturday, April 2, 3:15–3:35 Lane 124 A Model for Superior Risk-Adjusted Returns Using Statistics and Available Financial Tools The credit crisis of 2007–2008 reminded us all of the significant risk that financial markets present to investors. In particular, different asset classes tend to become more correlated during crises. This “black swan” weakens the value of traditional diversification. Quantitative financial models such as ours can significantly reduce this exposure. We discuss the statistical measures that support this conclusion and the statistical work we have done. Real money with proprietary data is used to provide a practical application of this theory with encouraging long-term results. A moderate amount of investment return is lost in applying this market neutral strategy (compared to more common strategies); however, the large reduction in risk has been an adequate trade-off. After years of application, the occasional pitfalls that statistics have revealed, along with corresponding model refinements, will also be discussed.
CLOSING SESSION
The closing session will include the award presentation for the MUMC and a plenary panel. Alissa Crans, Loyola Marymount University; Nina White, Uni- versity of Michigan Ann Arbor; and Yunus Zeytuncu, University of Michigan Dearborn Saturday April 2, 4:00–5:15 Dow Center A & B Math Circles and Math Teacher Circles in Michigan In this panel discussion we will explore what Math Circles and Math Teacher Circles are, how they are run, and the benefits for those organizing and those participating. Discussion will be driven primarily by audience qus- tions.
The Michigan Section acknowledges the sponsorship of
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