2015 Abstracts
Childhood Obesity
Claudy Eckardt, Weber State University Education Obesity is a growing epidemic in the United States. Statistical facts show that one third of U.S. children, between the ages of 2-19, are obese. A common health risk found in children who are overweight is that they will continue to stay overweight throughout the course of their lifespan. Long-term health consequences of obesity only to be found in adults have become more prevalent in children as well. Prevention is the key to reducing this dangerous epidemic and its consequences. The purpose of this study was to investigate children’s knowledge on the causes and consequences of being overweight. This study used a qualitative research method. Two participants were interviewed for case studies. Both participants were between the ages of 8 and 11 and were from different ethnic backgrounds. Each child was given nine questions to answer. Each child was given an adequate amount of time to thoroughly understand each question and respond. Interviews lasted between five to ten minutes. Parental consent was given before the interview process. Results showed that the participants were exposed to the risk factors of obesity. Each child demonstrated a clear understanding that obesity has negative health consequences and expressed preventative measures as well. Furthermore the participants proved that obesity was not only limited to the school or home environment but multiple environments.
Bold Talk for a One-Eyed Fat Man: The Importance of Classic Western Literature in the Contemporary Classroom
Are classic, young-adult novels set in the American west a dead genre in the American classroom? Despite its historical impact on popular culture, young-adult novels set in the American west have seen a considerable decline in relevance in American public schools. While classic young-adult literature receiving heavy rotation in current young-adult classrooms, as well as young-adult literature published in the current century, may have its place, Western, young-adult literature published in the 1960’s and 1970’s should still be read, taught, critiqued and celebrated. The general argument against including western, young-adult literature on the reading lists of public schools in the western states, where, arguably they should be the most prevalent, is that the texts are significantly passé and therefore not relevant to contemporary secondary students. While the majority of criticism in favor of western literature is also dated, this paper makes the argument that although the texts and various criticisms may seem behind the times, it should also be taken into account that present-day society is remarkably similar to society when the texts were initially published. Interestingly, not only is present-day society similar to society in the late 1960’s and 1970’s, but a recent interest in regionalism as opposed to vast commercialism also seems to be making an impact on society. By presenting literature that draws attention to the history of a local area, a literary gap created by an interest in who we were as a society and where we came from may be filled. The research conducted takes the classic, western, young-adult novel, True Grit by Charles Portis and that text with a more popular and more contemporary novel, The Outsiders by S.E. Hinton. It’s through this comparison we are able to appreciate and understand the relevance of classic Western literature in the contemporary classroom.
Content-Form Trade-offs in the Spontaneous Stories Told by Chi ldren with Autism Spectrum Disorder (ASD): Implications for Assessment and Instruction
Samantha DeLucchi, Telesha Fricke, Kamilla Okey and Brigid Crotty, Utah State University Education Children with ASD often experience marked difficulty achieving proficiency in narration, and often require explicit instruction to learn this important discourse skill. The present study was designed to extend the work of Colozzo et al., 2011 by examining the relationship between content and form in the narratives of school-age children with ASD as they participated in a narrative intervention program to improve their knowledge of story structure and ultimately, to improve their ability to create coherent, organized narratives. Children received two, individual, 50-minute intervention sessions weekly for a period of about 7 – 11 weeks. Children were asked to make up their own stories once weekly. These stories were scored for narrative proficiency and for grammatical accuracy. Findings revealed that prior to beginning narrative treatment, all of the children’s grammatical accuracy was high while their narrative proficiency scores were low. In the first weeks of treatment, all children experienced a significant decrease in grammatical accuracy (<70%), however their narrative scores were observed to increase. Narrative proficiency scores continued to increase and become stable for all children. Interestingly, grammatical accuracy returned to normal (90% or greater) during the last weeks of intervention as children’s narrative proficiency became stable. The findings from this study support the presence of a content-form tradeoff, as children learn difficult linguistic skills, other skills that are ordinarily stable, may fluctuate until the new skill is mastered. The absence of grammatical errors may not be taken as an indication that the student is proficient in constructing a coherent, organized narrative. Further implications are discussed.
The Relationship between Narrative Proficiency and Syntactic Complexity of Story Retells Elicited from Children with ASD
Taylor Anderson, Megan Israel Sen, Amy Nielsen, and Brigid Crotty, Utah State University Education Children with Autism Spectrum Disorders (ASD) have been shown to use substantially more simple sentences as compared to complex sentences in their spontaneously generated narratives (Sanford, Pearson, Summers, Crotty and Gillam, 2015). However, Sanford et al., showed that children with ASD began to use substantially more complex sentences in their stories during and after narrative treatment (>50%). It is possible that children with ASD may experience greater difficulty using complex sentences in stories they must generate than in retelling stories they have heard. The purpose of this study was to examine story retells of 5 children with ASD before, during and after narrative intervention for syntactic complexity. Results indicated that during baseline when children were not receiving instruction, their story retells contained more simple sentences than complex sentences. The use of complex sentences was observed to increase as children became more proficient in their narrative production skills. When compared to stories children generated on their own (spontaneous stories), the story retells contained more complex sentences overall, but were often associated with less verbal fluency particularly as children were mastering narrative skills. The findings will be discussed in terms of trade-offs in verbal fluency, grammaticality and the use of complex sentences during different stages of narrative proficiency as a function of initial language knowledge.
Characterization of the Role of Oxytocin in Larval Zebrafish Behavior
James Newton, Scott Anjewierden, and Sasha Luks-Morgan, University of Utah Engineering Oxytocin (OXT), a neuromodulatory peptide produced by the hypothalamus, is involved in a variety of physiological and behavioral phenomena. Exogenous OXT and drugs that mimic OXT signaling are potential treatments of a number of neurological disorders. The canonical mechanisms of OXT function are neuroendocrine in nature, as the peptide is released into circulation through the neurohypophysis. However, OXT has also been shown to exert some of its effects through direct synaptic release within the central nervous system. Using the larval zebrafish as a model, we seek to identify targets of these directly projecting OXT neurons and study what role they play in the modulation of behavior. Critical to this analysis are computer programs which enable precise quantification of anxiety, social behavior, and reward learning. Our custom-written software automatically identifies and tracks free- swimming fish, using measured positions over time to evaluate behavior in a variety of paradigms. In combination with molecular, cellular, and optogenetic manipulation of OXT networks, this project will allow a fuller understanding of the relationship between these neurons and behavior.
Methods for Simulating SAED and Kikuchi Diffraction Patterns in Atomistic Structures
Adam Herron, Jared Thomas, Shawn Coleman, Douglas Spearot, and Eric Homer, Brigham Young University Engineering For many years, x-ray diffraction and electron diffraction have served as effective means to understand and classify the molecular structure of many materials. Diffraction, as a physical phenomenon, is well known and theoretical diffraction simulation is relatively simple for perfect crystalline structures of known orientation. Prior methods of diffraction simulation, however, are insufficient to predict experimental diffraction patterns of unknown crystal structures or of crystal structures with high defect density. Recent advancements in computing capability and development of atomistic simulation software have greatly enhanced our ability to predict material properties and behaviors under various conditions. Atomistic simulation has become an extremely useful tool in the analysis of dynamic chemical and mechanical systems. It can only be truly effective, however, when it models a real-world application, can be interpreted coherently, and can accurately predict future conditions. Thus, we are developing new tools that bridge the gap between electron diffraction through real materials and simulated diffraction through atomistic simulations. We present a method of generating Kikuchi Diffraction Patterns from atomistic simulation data with no a priori knowledge of the crystal structure or crystallographic orientation. Our research was inspired by the recent work of Coleman et. al. 2013 and builds on their methods of calculating diffraction intensity at discrete locations in the reciprocal domain. We improve on their method by introducing an integration of the structure factor to ensure complete capture of diffraction intensity peaks while maintaining a relatively low density of sample points. This allows us to significantly reduce the required computation time on the analysis of atomistic simulation data. We use this diffraction data to generate simulated Kikuchi Diffraction Patterns.
In Vitro Cell-Free Synthetic Biology Techniques for Optimizing Protein Yields
Conner Earl, Brigham Young University Engineering The emerging field of Cell-free protein synthesis enables the efficient production of complex proteins for a number of exciting applications such as medicines that better interact with the body, vaccines, antibodies, and renewable, sustainable biocatalysts. However, progress is hampered by high costs and low yields of necessary proteins. This project is designed to improve protein yields and drive down costs by studying techniques of optimization of protein yields in Cell-Free protein synthesis. Our main area of focus is the inhibition of naturally occurring ribonucleases (RNAses) which are enzymes that degrade essential elements for protein synthesis- specifically, the mRNA used to transcribe protien. One of the techniques we intend to use for inhibition of these RNAses is by complexing the RNAse with an appropriate RNAse inhibitor protein thus limiting or eliminating its function of degrading mRNA. The aims of this research project is to: (1) Identify appropriate RNAse inhibitors (2) Design and synthesize inhibitor genes (3) Express, purify and assay RNAse inhibitors (4) Improve Cell-free protein synthesis yields utilizing RNAse inhibitors for analysis of activity and effectiveness as well as the enhancement of cell-free protein synthesis yields. Accomplishing these goals will result in more efficient systems and more accurate analysis that may lead to cheaper, more readily available vaccines and pharmaceuticals produced through Cell-free protein synthesis.
Purification of Air Using Molecular Modeling and Photocatalytic Nano-Materials
Nandini Deo, University of Utah Engineering Air quality in the United States has come under scrutiny in recent years. Many pollutants are trapped in the air we breathe in the form of photochemical smog. The aim of this research is to aid the breakdown of these pollutants. Peroxyacetyl Nitrate (PAN) is a predominant smog species; the research conducted aims to decompose this molecule and capture the resulting particles using the photocatalytic properties of Titanium Dioxide Nano tubes. The research conducted thus far has focused on the following questions:What molecules does the thermal decomposition of PAN produce? Is there a metal substrate to attach to TiO2 Nano-materials that aids the breakdown of PAN and its decomposition products? Can a sustainable process/device be identified to functionalize these materials? Literature research shows that PAN thermally decomposes into CO_2, NO_2, methyl nitrate, and formaldehyde. Methyl Nitrate and CO_2 may be eliminated using specific experimental conditions. Hence, it can be determined that the substrate attached to TiO2 must decompose PAN, NO_2 and formaldehyde. Using the molecular modeling programs Avogadro and MOPAC, 50 metals were optimized in relation to Formaldehyde, NO_2, and PAN. To find each metal’s reactivity to each target compound, HOMO/LUMO (Highest Occupied Molecular Orbital/Lowest Occupied Molecular Orbital) energies were calculated and used to find the common reactive metals between the target compounds: Cobalt, Silver, Iridium, and Niobium. To test whether the most complex product of the PAN decomposition (Formaldehyde) will break down, a device was created using a 3-D printer and Cobalt functionalized nanotubes. Pure formaldehyde, a blank sample (no tubes), and a sample with functionalized tubes were run through the device in the form of vapor, in front of a solar simulator. The captured vapor’s GC/MS results show an almost complete breakdown of Formaldehyde with the use of the device containing the functionalized tubes.
A Feasibility Study for Implementing a System of Electric Vehicles into Urban Environments
Carlton Reininger and John Salmon, Brigham Young University Engineering Electric Vehicles (EV) are a rising alternative to standard combustion vehicles because of their energy cost savings and reduced carbon emissions. However, EVs come with limitations such as limited driving range and potentially long recharge times. The purpose of this study is to determine the feasibility of implementing an electric vehicle system into an urban environment. Using data provided by the New York City Taxi and Limousine Commission, models are developed and generated to simulate driver shifts and analyze system level impacts from EVs on driver behavior. The models evaluate the number of charge events over the course of a shift and calculate the potential revenue lost to missed fares during charge intervals. Across the system, the results indicate that for a majority of NYC taxi drivers, EVs can be implemented without significant changes in driver behavior, while providing an economic and environmental advantage over current combustion vehicles. These preliminary findings can be used to support implementing such a system in urban environments and these models could be used as a template toward analyzing EV taxi potential in other cities.
Automated Kinematic Analysis of Prepulse Inhibition Behavior in Larval Zebrafish
Scott Anjewierden, James Newton and Joshua Barrios, University of Utah Engineering Organisms in their natural environment are constantly presented with sensory stimuli. These stimuli must be filtered by the brain to select an appropriate behavioral response. A significant example of this filtering process is audiomotor prepulse inhibition (PPI). In PPI, the startle response to a loud noise is suppressed by a preceding stimulus of lower intensity. This ability to optimize behavior in response to environmental context is an essential brain function. Defects in PPI are associated with neurological disorders such as obsessive- compulsive disorder, Tourette syndrome, and schizophrenia. This project demonstrates the development of new software to analyze swim kinematics in a restrained, larval zebrafish model of PPI. Our programs automatically extract several kinematic parameters from image sequences of behaving animals and use them to classify behavior into one of three, stereotyped categories. Correct classification is reported in 96.32% of trials (n = 162). This automated analysis will now permit a more robust study of PPI in these animals, where the brain’s experimental accessibility will allow us to discover the cellular bases of sensory filtering.
The Relationship between Trade Balance, Income and Real Exchange Rate
Wonjin Kim, Weber State University Business The trade balance of a country, calculated as total exports minus total imports, measures the impact of foreign trade on the demand for that country’s output. It is proposed that there are generally three key determinants of trade balance, e.g., the real exchange rate, home disposable income, and foreign disposable income. In principle, as the home country’s real exchange rate depreciates (rises), the country is expected to export more and import less. An increase in home country’s income is expected to increase imports and generate a decrease in home country’s trade balance. On the other hand, an increase in foreign income is expected to raise the home country’s trade balance. This paper makes an attempt to explore whether the expected relationships between trade balance, home income, foreign income and real exchange rate holds. The United States is considered the home country and Japan is considered the foreign country in this study. The reason for using Japan as a foreign country is twofold: first data availability and second, Japan is one of the largest trade partner of the United States. We use quarterly data to conduct our study. The sample period spans from 1994 to 2012. We hypothesize that the real exchange rate has a positive relationship with trade balance. Also, we hypothesize that the U.S. income has a negative and Japan’s income has a positive effect on the trade balance of the United States. To test our hypothesis we run a multiple regression. The dependent variable in our regression is trade balance. The independent variables are home and foreign incomes and real exchange rate.
Using Eye-tracking to Understand Professional First impressions in a P ersonal Selling Context
Josh Groves, Utah Valley University Business This study will discuss the contributions of eye-tracking data in testing first-time viewers of identified professional salespeople. Participants will view the professional portraits of two individuals, a male and female. With the results we hope to show that eye-movement data supplements what users verbally report in their reactions to an individual. In particular, eye-tracking data will reveal which aspects of the person receive more visual attention and in what order they are viewed. As eye-tracking systems have become more sophisticated and affordable, there has been an increasing interest in the use of eye-tracking within the first impression domain (Byrne, Anderson, Douglass, and Matessa, 1999; Kotval and Goldberg, 1998; and Hornof and Halverson, 2003). Despite increased research in this area, it is still not entirely clear exactly how much eye-movement contributes to the viewer’s evaluation of another individual. As part of a comparative test between two professionals’ images, participant eye-movements will be recorded during the initial introduction to the image. The eye-tracking data will be examined to gain additional insight into how users view that individual and what elements attract their attention. Additionally, respondents will be asked to provide their perceptions of five personality dimensions of the two individuals.
Bridging the GAAP between Accounting Education and Practice
Brock Griffith and David Lewis, Dixie State University Business In the past, some accounting researchers have criticized university accounting education programs. In their view, there is a gap between accounting education and practice. According to Sunder (2007), “The accounting curriculums of most business schools are full of theories that have less practical relevance.” The aim of this research is to identify the knowledge of subjects and skills needed for a relevant university accounting program. The views of over 200 professional accountants were sought and data collected using a questionnaire. In addition, several follow-up interviews and discussions were conducted with participants of the study. The findings suggest that the traditional accounting subjects are very important for a relevant accounting education. However, many professionals believe that certain aspects of accounting education are needed but currently the accounting education is not adequately meeting those needs.This paper helps to bridge the gap between academic study in accounting and a career in the professional practices. The results of the research will assist accounting educators in delivering work-ready graduates who will assist in meeting the needs of employers in the accounting profession.
Inflation and Unemployment in Greece
Qiheng Wu, Weber State University Business Research Question
Economical Rapid Production of Therapeutic Proteins using Cell -free Protein Synthesis
Hayley Ford, Kristen Wilding and Matt Schinn, Brigham Young University Engineering Therapeutic proteins are specially engineered proteins used to treat many large profile diseases. Such diseases include cancer, diabetes, hepatitis B/C, hemophilia, multiple sclerosis, and anemia. The use of these proteins is specific and highly successful and the demand for these proteins in rapidly increasing. One of the largest problems with the use of therapeutic proteins is the cost of making them. The cost of producing these proteins amounts to hundreds of billions of US dollars every year. There is a growing need to find better, faster, and cheaper ways to create them. As specific therapeutic proteins are coming off patent, research labs are able to explore the processes of making these drugs that have become such a large part of the pharmaceutical industry. Here we report the use of cell-free synthesis as a more cost-effective way to produce these therapeutic proteins. Cell-free protein synthesis is faster and allows for direct manipulation and control of the protein creating environment. Cell-free synthesis can produce proteins in a matter of days as opposed to the weeks it takes to produce them in vivo. The increased manipulation and control of the environment that comes with cell-free synthesis allows improved accuracy in creating the desired proteins and is more adaptable to changes if they need to be made.
Cardiac Tissue Engineering
Jordan Eatough, Jeremy Struk, Andrew Priest, Brady Vance, Brielle Woolsey, Steven Balls, Camille
Retinal Regeneration: Implications of Müller Cell Dedifferentiation
Theo Stoddard-Bennett and Steven Christiansen, Brigham Young University Engineering Damage to the human retina is often irreversible and so currently there are no established treatments of diseases such as dry age related macular degeneration (AMD). Dry AMD results in a loss of sight because of cell death in the macula, a centralized part of the retina which contains a high concentration of photoreceptor cells. One possible treatment would be to limit the rate of cell death within the macula, however this is not a comprehensive solution. Rather, regeneration of the photoreceptors within the retina is necessary to restore sight. In current research, Müller glia cells, a major glial component of the retina, can potentially be used as sources for photoreceptor regeneration in order to combat dry AMD due to their homeostatic regulation of retinal injury. Directed reprogramming would occur through a five step process. The Müller glia would need to undergo de-differentiation to Müller glia-derived progenitor cells (MGPCs), proliferation of MGPCs, migration of MGPCs, neuronal differentiation, and integration in order to generate retinal neurons. Müller cells can be isolated and cultured by dissociating retinal tissue in optimal media. Here we present the dissection and dissociation of rat retinal tissue to obtain purified proliferating Müller cell cultures. Our lab has tracked and modelled the rates of proliferation and phenotypically characterized the stages of proliferation. Using immunofluorescence and PCR tests to confirm purity, we will then expect to run a series of assays to identify growth factors, Wnt signals and cytokines to test the effects of retinal extracellular matrix proteins on Müller cell de-differentiation to MGPCs. The focus of our current research is the identification of reprogramming mechanisms that may possess beneficial data leading to both unique strategies for promoting retinal regeneration in mammals and clinical applications for those living with dry AMD.
Modeling Shale Oil Pyrolysis: Semi-empirical Approach
Dan Barfuss, Brigham Young University Engineering Shale oil has long been seen as a source of energy that can be incorporated into existing infrastructure. It consists of kerogen (or organic matrix) bound to inorganic rock. This kerogen can be released as an oil-like substance by heating it up to high temperatures without the presence of oxygen (i.e., pyrolysis). Due to advances in NMR (Nuclear Magnetic Resonance) we were able to make an accurate structural based model that can predict the relative tar and light gas yields[1]. We modified the Chemical Percolation Devolatilization Model (CPD) of coal to fit with the more aliphatic nature of oil shale. The CPD model describes the aromatic regions as clusters and aliphatic regions as bridges. As these bridges are broken the model releases groups of clusters that will form tar. In coal the bridge breaking gives off light gases, whereas in shale oil the bridges are much heavier and mostly form tar. We built two models that accounted for this. We also used the composition of the tar and the gas found by Fletcher et. al. [2] to predict what elements would be left and the aromaticity of the carbons. We found that throughout the reaction new aromatic regions were formed. With information from this model,- we are able to better predict the products of oil shale pyrolysis, and describe what happens chemically.
Manufacture of Hemocompatible Coronary Stents
Takami Kowalski, Warren Robison, Anton Bowden, and Brian Jensen, Brigham Young University Engineering Using a coronary stent to expand a blocked blood vessel as a way to treat coronary heart disease has proved effective in the past. However, there are risks, such as thrombosis, that are a natural side effect of inserting a foreign object into the body. Creating a stent out of a hemocompatible material such as carbon-infiltrated carbon nanotubes could potentially resolve these issues and also make unnecessary treatments such as dual antiplatelet therapy as a way of decreasing the risk of adverse side effects. Previous research done in this lab has shown that carbon-infiltrated carbon nanotubes can be grown in a pattern defined by photolithography on a planar surface. The present work demonstrates preliminary results from patterning a flat, flexible substrate and rolling it into a cylindrical shape before growing carbon-infiltrated carbon nanotubes as a way to fabricate cylindrical stents, fulfilling all necessary specifications for a stent with the added benefit of hemocompatibility. We also demonstrate growth on curved substrates and explore process parameters for achieving good-quality CNT forests.
Determining the Integrity of Decellularized Porcine Kidney Scaffolds
Benjamin Buttars, Jeffrey Nielson, Spencer Baker, Jonathon Thibaudeau, Angela Nakalembe, Tim
Expanding the Genetic Code Through Simultaneous Insertion of Unnatural Amino Acids
Steven Stanley, Brigham Young University Engineering The genetic code has long been restricted to a set of 20 fundamental building blocks called amino acids. Recent research has expanded the genetic code through unnatural amino acids (uAA), thus adding enormous possibilities to the potential chemistries of proteins. Because nature is highly selective in the protein translation process, it has proven extremely difficult to successfully insert multiple uAAs simultaneously. The incorporation of an uAA with in vitro methods typically relies on the use of the amber stop codon as a mutated insertion site. A stop codon placed at the middle of a gene can code for either the uAA or termination, thus, protein synthesis may often terminate prematurely instead of inserting the desired uAA. This inefficiency inhibits the possibility of inserting multiple uAAs simultaneously. We propose a novel method that will allow for multiple uAAs to be inserted simultaneously. Our method involves isolating a minimal set of tRNA for in vitro protein synthesis, allowing for uAA insertion to occur at codons other than the amber stop codon. My work has focused on the production of 4 versions of uAA-tRNA synthetase, a protein that charges tRNA with the uAA. We are currently growing these 4 different proteins in bulk and testing their activity. We will test them for compatibility, confirming that they do not interfere with one another and other synthetases native to our in vitro protein synthesis system. These uAA-tRNA synthetases, in conjunction with specialized tRNA, will provide the basis to efficiently incorporate multiple uAA simultaneously. The success of this project will have many practical applications ranging from new therapeutics to new methods of medical diagnosis.
The Musical Phiquence: Finding Phi in Musical Progressions
Sergio Arellano, Snow College Mathematical Sciences Why is music so pleasant to? In Western music, we have used what is called the “tempered scale” for centuries, and even though it has gone through changes, it is still largely based around the mathematical principles that Pythagoras created two thousand years ago. Is it possible that the explanation to this phenomenon has to do with the mathematical basis of music, which is unconsciously perceived by the brain? The human brain is known for detecting the underlying mathematical patterns present in many non-music related disciplines, such as visual arts. Independent of the music world, there is a proportion called the Golden Ratio or Phi. It is found in art and geometry, because human eyes tend to find that this proportion produces beauty. This is not a coincidence; this proportion is found everywhere around us, especially in nature. With this in mind, it makes sense to search for the Golden Ratio in the tempered scale to explain the mind’s intuitive appreciation of music. This research discovered the surprising fact that there is a Phi relationship between the first and third degree of the major scale, in terms of the frequency of sounds. The relationship is this: the sum of the frequencies of the notes of the minor III chord divided by the sum of the frequencies of the major I chord in any given major key, tends to be Phi. Another very surprising result was the importance of the number 24 in the frequencies of the major scale. These two particularities help to shed light on why human ears have an untaught comprehension of music; the underlying perception of mathematical relationships by the mind are related to the natural appreciation of it.
Study the Stability of Steady Solutions for a Model of Mutualism
Amy Gifford and Brennon Bauer, Southern Utah University Mathematical Sciences Mutualism is the way two organisms of different species exist in a relationship in which each individual benefits from the activity of the other. We study a mathematical model of mutualism. The stability of the steady state solutions of this system will be analyzed. Also, we give some numerical experiments that verify the theoretical results for those steady solutions.
Counting Integer Points in Scaled Polytopes
Christopher Vander Wilt and Daniel Gulbrandsen Utah Valley University Let nP denote the polytope obtained by expanding the convex integral polytope P⊂R^d by a factor of n in each dimension. Ehrhart [1] proved that the number of lattice (integer) points contained in nP is a rational polynomial of degree d in n. What happens if the polytope is expanded by not necessarily the same factor in each dimension? In this talk a partial answer to this question will be provided, using powers of n as different factors to expand the polytope. It will be shown that the number of lattice points contained in the polytope formed by expanding P by multiplying each vertex coordinate by such a factor is a quasi-polynomial in n. Quasi-polynomials are a generalization of polynomials, where the coefficients of the quasi-polynomials are periodic functions with integral period. Furthermore, particular cases where the number of such lattice points is a polynomial will be presented. In addition, the period of these quasi-polynomials as well as the Law of Reciprocity will be addressed. At the end, future work will be discussed. [1] E. Ehrhart, “Sur les Poly`edres Rationnels Homoth ´etiques `a n Dimensions,” C. R. Acad. Sci. Paris 254 (1962)
The Divisibility of p^(n)-1 for p>5, p a Prime Number
Jason Adams, Nathan Jewkes and Tyrell Vance, Southern Utah University Mathematical Sciences We will study the divisibility of p^(n)-1 where p is a prime number larger than 5 and n is a positive integer. We will generalize the result by considering the case where n is odd and two cases where n is even. We show that when n=2^(k), k an integer greater than 1, 2^(k+2)∙3∙5 is a factor of p^(n)-1. We also show that when n=2^(m)∙l for m a positive integer greater than 1 and l an odd positive integer greater than 1, 2^(m+2)∙3∙5 is a factor of p^(n)-1.