MCM2018优秀论文.pdf

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Wave Goodbye to Poor Reception

Summary

Introduction

Problem Summary

Our Starting Point: Existing Models

Our Model

Background

Transmitter and Receiver

Radio Propagation and Loss

Skywave Channel Interaction

Ionosphere Interaction

Surface Interaction

Performance Evaluation

Assumptions

Model Assumptions

Nomenclature

Model Development

Model Construction

Propagation and Ionosphere Interaction

Ground Interaction

Loss

Model Validation

Model Application

Part I

Part II

Part III

Results

Validation

Part I

Part II

Part III

Sensitivity Analysis

Explore Parameter Space

Frequency and Elevation Angle

Wave Turbulence

Transmit Power

Ionosphere Density

Conclusion

Our Conclusion

Strengths

Weaknesses and Limiting Assumptions

Letter

References

About the Authors

The UMAPJournal 93 . FL 32547 jmcargal@gmail.com rd

2018 Vol. 39, No. 3 Table of Contents Publisher’s Editorial What Good Are Contests? Solomon A. Garfunkel ...................................................................237 Editor’s Note Paul J. Campbell ............................................................................240 MCM Modeling Forum Results of the 2018 Mathematical Contest in Modeling Patrick J. Driscoll ...........................................................................241 Wave Goodbye to Poor Reception Marc Thomson, Derek Gorthy, and Christine Reilly .........................263 Judges’ Commentary: Multi-Hop HF Radio Propagation Joanna Bieri and Kelly Black...........................................................287 How Many Languages? Jiqun Chu, Jinfei Zhu, and Lingyun Feng ........................................295 Judges’ Commentary: How Many Languages? William P. Fox and Robert E. Burks .................................................313 Better to Marry Renewables than to Burn Fossil Fuels in Border States Aimee Maurais and Arianna Krinos................................................321 Judges’ Commentary: The Southwest States’ Energy Compact Katie Oliveras, Stacey Hancock, and David H. Olwell ......................343

Publisher’s Editorial 237 Publisher’s Editorial What Good Are Contests? Solomon A. Garfunkel Executive Director COMAP, Inc. 175 Middlesex Turnpike, Suite 3B Bedford , MA 01730–1459 s.garfunkel@comap.com Introduction Since 1985, we have run modeling contests. In fact, COMAP now ad- ministers the MCM R/ICM R contests at the undergraduate level as well as HiMCM R and IMMC R at the high school levels All told, more than 23,000 teams and well over 60,000 students competed this past year in one or more of these. COMAP is clearly in the contest business! In what ways does this make sense? After all, we do not believe that mathematical modeling is a competitive sport, and we certainly do not wish to promote it as such. Our original proposal to the Department of Ed- ucation’s Fund for the Improvement of Post-Secondary Education (FIPSE), and every public statement we have made since, all clearly state that the purpose of our contests is to promote the teaching and learning of mathematical modeling and applications. After all these years, it’s fair to ask whether this is working, or— more generally—what has been the result of all of this work? Serendipity in Teamwork I should begin by mentioning a completely unplanned effect. A num- ber of years ago, we began collecting data on the gender of our contest participants. Figure 1 shows those results from 2010 to the present. We note two things: The UMAP Journal 39 (3) (2018) 237–240. cCopyright 2018 by COMAP, Inc. All rights reserved. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for proﬁt or commercial advantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP.

238 The UMAP Journal 39.3 (2018) Male vs. Female Students 35000 30000 25000 20000 15000 10000 5000 0 0 1 0 2 1 1 0 2 2 1 0 2 3 1 0 2 4 1 0 2 5 1 0 2 6 1 0 2 7 1 0 2 8 1 0 2 Figure 1. Male (blue, on left) and female (pink, on right) participants in the MCM/ICM from 2010 through 2018. Female Male • the percentage of female participants has increased each year since we began collecting the data, and • the percentage of women on the Outstanding teams in 2018 is precisely the same percentage (43%) as that of all female participants. This year, COMAP and Stanford University jointly funded a study of this phenomenon with an eye to understanding why female participation in COMAP contests is so vastly different from that on more standard math- ematics contests. The work, conducted by Prof. Jo Boaler and her team, was reported on at Mathfest in Denver in August 2018 and will be reported on again at the Joint Mathematics Meetings in Baltimore in January 2019. There are clearly a number of aspects of the contests that are attractive to women; but by far the most important is the collaborative aspect of our contests: teamwork, pure and simple. Again, in all of our literature, we have made the point that in the real world people work together to solve problems. It is only in mathematics classes that we seem to make a point of restricting communication. Moreover, as in our contests, in the real world results have to be communicated to others, often in lay terms. Are We There Yet? But what of our initial goal? Have we in fact helped to increase the presence of mathematical modeling in schools across the country? I sus- pect that the answer here is yes, with a caveat. Certainly, there are more modeling courses today than in 1985. Certainly, a number of schools have instituted seminars to prepare students for MCM/ICM. And it is not an accident that the Common Core State Standards in Mathematics included mathematical modeling as a basic mathematical practice. But modeling still has an uphill battle to become a major part of the K-12 mathematics curriculum. In the spirit of unintended consequences, we have had a much greater impact on Chinese education than on that in our own country. The Chi-

Publisher’s Editorial 239 nese have truly adopted modeling as a major part of their mathematics education, even to the point of having high schools and universities with “modeling” in their names. Their high-stakes test has changed to add performance-based modeling problems, and their K-12 curriculum is chang- ing to reﬂect this addition. This effort is strongly supported not just by the Ministry of Education, but also by local industry anxious for employees with modeling skills. The Contests Make a Difference So, we have a long way to go to meet the goals that we set for our- selves as we began MCM 34 years ago. But when I look at the underlying question of what we have accomplished, I can’t help but reﬂect on the com- ments that I have heard over and over again by people who participated in one or more of these contests as high school or undergraduate students. People come up to me at math meetings or write emails that say in very stark terms: “This contest changed my life.” They tell of ﬁnally understanding how mathematics can be used to change the world and how they could be part of that effort. They see a different future for themselves and a growing conﬁdence that they belong. Again, the team experience is emphasized. But more than that, they speak of freedom—the freedom to make their own judgments on how to attack a problem, or what data makes sense to use—a freedom that they had never experienced in a mathematics class. So while our ultimate goal is to have all students have such experi- ences, and to change the mathematical education of all students, I believe that these competitions have done some good and made a real difference in real lives. We can do more, and we will keep trying to spread the word. It is important that we always keep in mind that we strive to have stu- dents better understand mathematics and the role it plays in their world. Contests are a surprisingly effective tool, but nonetheless a tool to an end beyond themselves. About the Author Solomon Garfunkel is the founder and Executive Director of COMAP and Executive Publisher of this Journal. He served on the mathematics faculties of Cornell University and the University of Connecticut at Storrs, but he has dedicated the last 35 years to research and development efforts in mathematics education. He was project director for the Undergraduate Mathematics and Its Applications (UMAP) and the High School Mathematics and Its Applications (HiMAP) Projects funded by NSF, and directed three telecourse projects, including

240 The UMAP Journal 39.3 (2018) Against All Odds: Inside Statistics and In Simplest Terms: College Algebra, for the Annenberg/CPB Project. He has been the Executive Director of COMAP, Inc. since its inception in 1980. Dr. Garfunkel was the project director and host for the video series For All Practical Purposes: Introduction to Contemporary Mathematics. He was the Co-Principal Investigator on the ARISE Project, and Co-Principal Investi- gator of the CourseMap, ResourceMap, and WorkMap projects. In 2003, Dr. Garfunkel was Chair of the National Academy of Sciences and Math- ematical Sciences Education Board Committee on the Preparation of High School Teachers. He was the recipient of the National Council of Supervi- sors of Mathematics Glenn Gilbert Award. He served on the Mathematics Expert Group for PISA 2012. Editor’s Note The views and opinions expressed in this issue by authors employed by the U.S. Department of Defense are theirs alone and not necessarily those of the Department of Defense or of any agency of the U.S. government.

Results of the 2018 MCM 241 MCM Modeling Forum Results of the 2018 Mathematical Contest in Modeling Patrick J. Driscoll, MCM Director United States Military Academy West Point, NY chi.decisions@gmail.com Introduction A total of 10,670 teams of undergraduates from hundreds of institutions and departments in 17 countries/regions spent a weekend working on ap- plied mathematics problems in the 34th Mathematical Contest in Modeling (MCM) R. The 2018 MCM began at 8:00 P.M. EST on Thursday, January 19, and ended at 8:00 P.M. EST on Monday, Januuary 23. During that time, teams of up to three undergraduates researched, modeled, and submitted a so- lution to one of two open-ended modeling problems. Students registered, obtained contest materials, downloaded the problems and data, and en- tered completion data through COMAP’s MCM Website. After a weekend of hard work, solution papers were sent to COMAP on Monday. Three of the top papers appear in this issue of The UMAP Journal, together with commentaries from the contest judges. The Interdisciplinary Contest in Modeling (ICM) R, a companion con- test, took place concurrently over the same weekend. The ICM offers mod- eling problems involving network science, human-environment interac- tions, and policy modeling. Details about the 2018 ICM Contest and its results are in Vol. 39, No. 2 of this Journal. The 2019 MCM/ICM Contests will take place January 24–28, 2019. The UMAP Journal 39 (3) (2018) 241–261. cCopyright 2018 by COMAP, Inc. All rights reserved. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for proﬁt or commercial advantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP.

242 The UMAP Journal 39.3 (2018) Description of the Problems This year, the three MCM problems represented interesting scenarios for contestants, each offering a dimension of mathematical modeling that was unique. The authors of the problems were Dr. Michael Tortorella (Rut- gers University) (Problems A and B) and Dr. Kelly Black (Clarkson Univer- sity) (Problem C). All of the competing teams are to be congratulated for their excellent work and enthusiasm for mathematical modeling and interdisciplinary problem solving. Resources for Mathematical Modeling COMAP, whose educational philosophy is centered on mathematical modeling, supports the use of mathematical concepts, methods, and tools to explore real-world problems. COMAP serves society by developing stu- dents as problem solvers in order to become better informed and prepared as citizens, contributors, consumers, workers, and community leaders. The MCM is an example of COMAP’s efforts toward these goals. COMAP’s Mathematical Contest in Modeling and its Interdisciplinary Contest in Modeling are the only international modeling contests in which students work in teams. In addition to this special issue of The UMAP Journal, COMAP offers at www.mathmodels.org the press releases for the 2017 contests, their results, their problems, unabridged versions of all the Outstanding papers, and judges’ commentaries. Results and winning papers from previous contests were published in special issues of Mathematical Modeling (1985–1987) and The UMAP Journal (1985–2017). The 1994 volume of Tools for Teaching, commemorating the tenth anniversary of the contest, contains the 20 problems used in the ﬁrst 10 years of the contest and an Outstanding paper for each year. That vol- ume and the special MCM issues of the Journal for the last few years are available from COMAP. The 1994 volume is also available on COMAP’s special Modeling Resource CD-ROM. Also available is The MCM at 21 CD- ROM, which contains the 20 problems from the second 10 years of the con- test, an Outstanding paper from each year, and advice from advisors of Outstanding teams. These CD-ROMs can be ordered from COMAP at http://www.comap.com/product/cdrom/index.html . Contest problems and results of the MCM/ICM contests are on the COMAP Website at http://www.comap.com/undergraduate/contests . The volume Mathematical Modeling for the MCM/ICM Contests Volume 1 exposits the ideas, background knowledge, and modeling methodologies

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