BRIDGING THE GAP: OBSTACLES AND OPPORTUNITIES FOR KNOWLEDGE TRANSFER IN EVIDENCE-BASED MANAGEMENT

 

Flore Ravaud (Dr. Aikaterini Manthiou, Department of Marketing - NEOMA Business School & Dr. Lawrence Garber) Department of Marketing and Entrepreneurship 

Visual merchandising plays a crucial role in customer satisfaction and sales performance. Many practitioners follow a set of rules for their shelf management: eye-level, symmetry, repetition, color blocking, etc. However most of those golden rules have never been proven scientifically. The purpose of this study is to scientifically test the visual merchandising practices globally adopted by retailers. I propose an experiment to test the validity of practitioner’s rules of thumbs on visual merchandising on how to optimize shelf management using A/B testing. An experimental novel method for their testing is proposed, and expected results are discussed. 

 

Sabina Bains (Dr. Kirsten Doehler) Department of Mathematics and Statistics  

Sports analytics is growing in popularity, but research related to distance running is sparse.  One aim of this study is to examine finishing times and pacing strategies of participants in the 5-Kilometer (5K) and 10-Kilometer (10K) races at the NCAA Division 1 Outdoor Track & Field Championships from the years 1998 to 2015.  We also investigate the variability in finishing times to examine whether this was correlated with the winning time. It was determined that finish times of female athletes had a higher variability than those of male athletes. In the 5K races the variability in finish times tends to be smaller when then winning time in that year is slower.  However, this trend is not observed in the 10K races.  Additionally, we investigate the participation rates of different Division 1 programs and provide our opinion on which colleges have exemplary distance running programs.  Stanford had 111 athletes (6.9% of all athletes) qualify for either the 5K or 10K race from 1998 to 2015, which is significantly more than any other school.  Data for this study was obtained from the flashresults.com website, which has information on NCAA Championship race times from 1998 to 2015.   

 

Taylor C. Cesarski (Dr. Chad Awtrey) Department of Mathematics 

Finding solutions of polynomial equations is a central problem in mathematics. Of historical importance is the ability to solve a polynomial "by radicals''; i.e., using only the coefficients of the polynomial, the four basic arithmetic operations (addition, subtraction, multiplication, division), and radicals (square roots, cube roots, etc.). Polynomials of degree four or less have been shown to be solvable by radicals, while the same is not true for higher degrees. How do we determine which polynomials are solvable by radicals? To answer this question, we study an important object that is associated with every polynomial. This object, named after 19th century mathematician Evariste Galois, is known as the polynomial's Galois group. The characteristics of the Galois group encode arithmetic symmetries of the corresponding polynomial's roots, and these symmetries can determine whether or not the polynomial is solvable by radicals.  We discuss a new algorithm for determining the Galois group of a degree seven polynomial which improves upon prior research.  In particular, previous methods rely on factoring two or more auxiliary polynomials while ours requires only one. 

 

Nicole B. Ciotoli and Jennifer L. Faig (Dr. Todd Lee) Department of Mathematics and Statistics 

This scholarship of learning sought to find a new and appealing way to utilize the game of SET (a specialized deck of cards) to engage people in the complex mathematical concept of an affine finite geometry through a video presentation. Understanding an affine finite geometry requires the grasping of ideas including axiomatic constructions, geometric models, and algebraic fields. It is a growing trend for YouTube speakers to discuss complex STEM concepts in similar ways using short, well-crafted videos, like the one created in this project. Papers on the topic of SET, and the math behind the game, were used to gain an understanding of the theoretical principles necessary for such a presentation. SET is a math game, which contains cards of varying attributes, including color, shape, number, and shading. The goal of the game is to find as many SETs as possible from a certain number of cards from the deck (a SET being three cards where each attribute is the same for all three cards or different for all three). Exploratory work was conducted using the SET cards to fully understand the geometries the cards were capable of modeling. The exploratory phase concluded with a finite geometry, modeled using all 81 cards in the deck, and containing thousands of different sets and hypercubes. It was verified that this final geometry satisfies the required axioms in order to be considered a model of an affine geometry. By assigning vector representations for each card and using linear algebra, or SET logic, the relationships between the initial cards and the other cards in the figure can be solved for in order to incorporate all 81 cards into the geometry. Both the linear algebra approach and the approach using SET logic are used in our demonstration, with explanation of how the two are equivalent. The final product is a comprehensive video, which utilizes animations, mathematical diagrams, and the actual SET cards to explain the concept of an affine finite geometry. 

 

Jennifer L. Faig and Jessica M. Weiss (Dr. Kirsten Doehler) Department of Mathematics and Statistics 

Marathon running is a global athletic event that millions of people take part in.  Women were not allowed to officially enter a marathon race until the New York City Marathon opened participation to women in 1971 and the Boston Marathon allowed women to run in 1972.   At first women’s finishing times in marathons seemed to be getting closer to men’s finishing times.  Controversy arose regarding gender differences in marathon running and this inspired many research studies.  The purpose of our study was to analyze numerous years of data, up to and including data from 2014, in order to generate an idea of trends over time and current gender differences in competitive marathon running. These analyses helped us predict whether it is likely that women will ever outrun men during a traditional marathon distance. We considered race times from the Bank of America Chicago Marathon from 2000 to 2014. In order to look at the relationship between finish times of each year and gender, a regression analysis was performed using results from the first place male and female finishers and the top ten male and female finishers in each race. Furthermore, numerous tables were generated to show average split times, finishing times, and pacing for each age group and gender. We also examined differences in split time between the first and second half of the marathon for all finishers.  These analyses allowed for general conclusions about the running community and the possibility of women outrunning men. Results indicate that there is a significant difference in the true mean finish time between males and females for all age groups except 80 years and older. It was also discovered that, on average, men tend to slow down more in the marathon than women. 

 

Eric J. Goding (Dr. Crista Arangala) Department of Mathematics & Statistics 

Advertising analysis can be an important tool for social scientists who wish to understand values in a society because successful advertisements reflect these values. Researchers comparing based on their values can find it useful to examine attributes of advertisements. Qualitative content analysis is the most common method to compare advertisements cross-culturally or cross-generationally. However, quantitative methods, mainly chi-square tests, can also be used. These tests compare two countries based on one attribute, where the attributes are defined as present or absent within an ad. The result of the test gives evidence for a difference or no difference between the two countries. In our research, we replicate this process, but we also seek to compare more than two countries at a time using linear algebra techniques such as seriation and single-value decomposition. These techniques place countries within the rows of a binary matrix and attributes within its columns, with 1s or 0s in the cells depending on whether or not a country’s ads possess an attribute. Countries are reordered based on dissimilarity between the rows (countries) of the matrix. The output is a ranking where similar countries are close together and dissimilar countries are far apart. The output given by these methods matches output given by the chi-square testing, which supports that these methods are valid for comparing different countries’ advertisements. This research could provide new mathematical techniques to compare several countries’ advertisements and further understand which countries are similar and different. 

 

Alexandra N. Horowitz (Dr. Laura Taylor) Department of Mathematics and Statistics 

Using statistical analysis is valuable to almost every academic discipline when analyzing research data. Although most college majors do not have statistics requirements, many academic departments use statistics in research analysis, and students could benefit from an increased knowledge of statistical concepts. The purpose of this research was to investigate the use of statistics within past SURF projects. An online survey was sent to SURF participants in both 2014 and 2015 with various demographic, research based, and general statistics knowledge questions. If students responded that they had used statistical analysis for their research project, they were asked a series of questions regarding the types of analysis they performed, what outside resources were utilized, and the percentage of analysis done by their mentor. Additionally, the students were asked about their perception of the value statistics added to their research findings. For those who did not use statistical analysis, they were asked whether or not they believe statistical inference could have added value to their results. All students who chose to participate in the study were asked about the definition of a p-value. The results of this questionnaire showed that most SURF presenters who used statistics for their projects came from either STEM or social sciences disciplines. The majority of students who conducted research for SURF had taken at least one college level introductory statistics course, and were able to use a variety of types of data analysis. Although most respondents had taken at least an introductory statistics course, there were extremely mixed results on their ability to define a p-value. Only about 40% of respondents were able to choose a correct definition given two correct and three incorrect choices. The main results of this study show that although a large amount of students used statistical analysis to add to their research findings, there is still much room to improve in terms of increasing general statistics knowledge among students within all academic disciplines.  

 

Michael R. Keenan (Dr. Chad Awtrey) Department of Mathematics and Statistics 

In the 1500s mathematicians discovered that all quartic polynomials are solvable by radicals, meaning we can find a quartic polynomial's roots using only the coefficients of the polynomial, the basic arithmetic functions (addition, subtraction, multiplication, and division), and radicals  (square roots, cube roots, etc.). It wasn't until the 1800s when mathematicians showed why quartic polynomials are solvable by radicals and why not all polynomials of degree greater than four are. By attaching a group structure to a polynomial (called the polynomial's Galois group), we can determine whether the polynomial is solvable by radicals.  We can also see the relationships among the roots. Naturally, a branch of mathematical research has emerged to develop methods to determine Galois groups of polynomials. Previous methods for determining Galois groups of quartic polynomials have involved factoring and creating larger polynomials (called resolvent polynomials); a process that can be computationally inefficient. The first aim of our research is to propose a new method of computing the Galois group of a quartic polynomial that does not rely on factoring large-degree resolvents.  Instead, we use only two pieces of data about the polynomial: (1) the number of roots in the field extension it defines, and (2) its discriminant. Our second goal is to compare the efficiencies of the previous resolvent-based methods with our new method. By coding an algorithm for each method into Wolfram Mathematica and measuring the time it takes for each algorithm to compute the Galois Group of quartic polynomials of various sizes, we are able to compare the efficiencies of each method. We find that our new method is not as efficient as other resolvent-based methods. 

 

Stephanie M. Lobaugh (Dr. Laura Taylor) Department of Mathematics and Statistics 

When investigators aim to compare the population means of two or more populations, an inferential method known as analysis of variance (ANOVA) is employed.  Experiments that use ANOVA analysis are common in social and biological sciences.  When investigators perform ANOVA, they must verify that their data satisfies several assumptions in order to ensure the validity of any conclusions drawn from the analysis.  One assumption requires that the population variances be equal; the populations must have equal distribution of data around their respective population means.  The purpose of this study is to evaluate the effectiveness of various rules of thumb used to check the assumption of equal variances for ANOVA. There is evidence of a general lack in checking the assumption of equal variance; this disregard can lead to an unacknowledged violation, which could profoundly impact the validity of ANOVA conclusions.  Such impact could result in Type I Error, or the false conclusion that at least one population mean is different.  This potential consequence demonstrates the significance of checking the assumption and, by extension, the importance of evaluating simple rules of thumb that are related to the ratio of the sample variances and used to verify the assumption.  The drive for this investigation stems from the existence of several variations on the rule of thumb presented in university-level textbooks.  This study aims to provide researchers with advice regarding assumption verification to ensure the validity of their conclusions derived from ANOVA.  Simulations were run in R in which samples were drawn from multiple populations with known variances. An ANOVA F-test was performed using these samples and the frequency of Type I Error was observed in order to quantify the impact of the violation.  The effectiveness of each rule of thumb is discussed based on the frequency of the observed Type I Error rates. 

 

Nathan M. Pool (Dr. Jeff Clark) Department of Mathematics 

Have you ever gazed into a work of art, a coastline on a map, or even an aspect of nature like a leaf or a snowflake and noticed a repetitive pattern the closer that you observe it? Figures that have this quality, known as self-similarity, are considered fractals. Mathematically, these shapes transcend traditional dimension – dimension being a measure of visual complexity. An unlikely but interesting connection can be drawn between fractals and musical composition when considering the idea of self-similarity. Classical composers such as Beethoven and Bach composed their works utilizing self-similarity to establish themes in their music. This is where the idea of fractal music comes into play. Automated musical composition is the method of using mathematics to mimic music written by human composers. Fractal music is a specific branch of automated musical composition that focuses solely on recreating the self-similar, thematic, repetitive aspect of music. This investigation, never documented in research literature, maps the coordinates of fractals of designated dimension to musical notes by constructing algorithms in Wolfram Mathematica. Because the dimension of fractals is so variant depending on their construction, the project examines the correlation between the dimension of each fractal and the corresponding sounds. Specific correlations between the two are still to be found, but it does seem after examining the sounds of each fractal that compositions corresponding to larger dimension tend to be more melodically sporadic than compositions of smaller dimension. The purpose of fractal music and automated musical composition is to investigate the parallels between mathematical computation and music. What is the nature of musical composition and songwriting? Is it purely mathematical or are there some aspects of music that mathematics and computation cannot predict and recreate? While these questions are still left unanswered, there is vast potential for trends to be found between the behavior of fractal music compositions and their corresponding fractal dimensions. This is one step closer to more accurately imitating the human thought process in musical composition.