SPECTRUM Journal of Student Research at Saint Francis University
Volume 1 Spring 2011
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SPECTRUM: A Journal of Student Research at Saint Francis University Faculty Editors:
Balazs Hargittai Associate Professor of Chemistry bhargittai@francis.edu
Student Editorial Board:
Jenna Bailey Seth Burkert Sean Gdula Lindsay Jerin Jonathan Miller ’08 Aaron Osysko ’10 Macy Rupprecht Colleen Stock
Managing Designer:
Brittany Kovacs
Grant Julin Assistant Professor of Philosophy gjulin@francis.edu
Rachel Blinn Emily Coy Eric Horell Paul Johns ’07 Sean O’Reilly Darren Petrunak Jennifer Sabol Amanda Young
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SPECTRUM
Journal of Student Research at Saint Francis University Volume 1 Spring 2011
Table of Contents Editorial
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Letter from the Provost
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Long-Term Recession Forecasting Using the Yield Curve. Jennifer M. Sabol; Edward J. Timmons
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An Invariant of Metric Spaces Under Bornologous Equivalences. Brittany H. Miller; Laura M. Stibich; Julia H. Moore; Brendon LaBuz
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Running into the Unknown: The Religious Dimensions of Distance Running. Andrew R. McKee; Arthur Remillard
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Call for papers
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(Student authors’ names underlined.)
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Editorial Student research is becoming an important part of the education of an increasing number of Saint Francis University students. Presentations at Undergraduate Research conferences have shown that participating in student research enhances students’ college experiences and benefits them by improving their knowledge in their chosen disciplines, their writing and verbal skills, and their self-confidence. Student research also enhances students’ problem solving skills, their critical thinking abilities and better prepares them for professional practice or graduate school. SPECTRUM is a peer-reviewed undergraduate research journal featuring the works of Saint Francis University students. The mission of this journal is to report on and disseminate the results of Saint Francis undergraduate research to those within and outside our academic community. This journal is not to be considered a substitute for established undergraduate research publications. However, should undergraduate scholarship at our University become published in a preeminent journal, SPECTRUM would hope to publish summaries of said research so that we may share the achievements of undergraduate research with our academic community. In addition, our mission serves to familiarize our contributors with the process of preparing research reports, corresponding with journal editors, and the overall practice of academic publication. SPECTRUM was launched to showcase the work of students who have become engaged in research at our University. Once (or twice, depending on submitted papers) a year we will publish articles written by our University’s students from across all disciplines. Our Student Editorial Board is made up of creative and driven undergraduate students, who come from diverse backgrounds such as Accounting, Biology, and English. We are looking forward to receiving contributions from as many students as possible and call on both undergraduate and graduate students at Saint Francis University who are involved in student research to share their research with the University community through SPECTRUM. Balazs Hargittai and Grant Julin, Faculty Editors
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Letter from the Provost Welcome to the inaugural issue of SPECTRUM, the Saint Francis University scholarship journal. All those involved in its production are thrilled to have the dream become a reality. Scholarship is playing an ever more importantly role as a part of the university curriculum. Our faculty have long striven to maintain their contribution to the development of their profession and they have always recognized their responsibility to expose students to that scholarship and to original resource material. What has changed is the expectation and level of involvement of students in scholarship. Our students now routinely help faculty with their efforts as well as conceive, execute, and report on projects of their own. The benefits of student-faculty partnerships extend beyond the classroom. Students gain a firsthand appreciation of how the discipline grows and changes while they hone their ability to think critically and creatively. Faculty, in turn, are invigorated as they gain a new set of colleagues that challenge their ideas and their methods. And, the university itself benefits as it takes its place amongst those institutions that create new knowledge and ideas rather than just consuming them. Thank you for picking up this issue of SPECTRUM. Celebrate with us the accomplishments of a very talented group of Saint Francis University students.
Wayne Powel
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L ong-T er m R ecession For ecast ing Using t he Y ield C ur v e Edward J. Timmons, Ph. D. School of Business etimmons@francis.edu
Jennifer M. Sabol School of Business jmsst36@mail.francis.edu
In this paper, we examine the accuracy of long-term (6- and 8-quarter) recession forecasting using the yield curve. We specifically focus on the yield spread, which is the difference in interest rates between a 10-year US Treasury note and a 3-month US Treasury bill. We compare the economic forecasts from the yield curve to the forecasts made by the Survey of Professional Forecasters (SPF). Our results suggest that although some degree of accuracy is lost, the yield spread outperforms forecasts from the SPF made over shorter horizons. The yield curve seems to provide some insight into the state of the United States’ economy as far as two years into the future. Keywords: yield curve, yield spread, forecasting, recession Pr eface Causal observers of the yield curve may not be cognizant of its importance. The purpose of this paper is to enlighten readers on why the yield curve is important. By using the yield curve, readers lacking economic knowledge can gain some insight into the future direction of the United States economy (as measured by gross domestic product or GDP). This paper touches briefly on the history of the yield curve, but more closely delves into the following question: how far in advance can Americans know there is a recession coming in the future? Most research has focused on a year or less into the future; however, this paper looks specifically at one and a half years and two years into the future. If Americans, as well as American companies, can know a recession is coming two years ahead of time, they can prepare, save now for the future, and possibly avoid a devastating financial crisis. Int r od uct ion In the existing literature, there is evidence supporting that the difference between the yield of 10-year Treasury bonds and the yield of 3-month Treasury bills (or “the yield spread”) can be used to forecast economic conditions in future quarters.
Estrella and Mishkin (1996) showed that the yield curve (the graphical plot of the relationship between the yields on bonds which only differ in maturity length) can be a valuable forecasting tool in predicting recessions two, four, and six quarters in the future using thirty-five years of data, starting from the first quarter of 1960 and finishing with the first quarter of 1995. They utilize a probit model from out-of-sample forecasts that estimates the probability of recession as a function of the yield spread. Their results suggest that the yield spread outperforms the New York Stock Exchange stock price index, the Commerce Department’s index of leading economic indicators, and the Stock and Watson index when predicting recessions two or more quarters in the future. In a more recent paper, Estrella (2005) states, “the yield curve has predicted every recession since 1950 with only one false signal, which preceded the credit crunch and slowdown in production in 1967” (p. 2). In our study, we investigate the ability of the yield spread to predict recessions further into the future than previous studies. If the yield spread’s forecasting ability is accurate up to two years in the future, more attention should be paid to signals from the yield spread. We rely on data from the second quarter of 1953 to the third quarter of 2009,
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a span of 56 years and one quarter. We follow Estrella and Mishkin (1996) by using a probit model, an econometric forecasting tool, to estimate the probability of recessions. As a side note, a probit model is used for a relationship between one discrete response variable and one or more independent variables in place of a linear probability model. The goal of a probit model is to transform a curve into a linear relationship that can then be analyzed using ordinary least squares (OLS) regression or maximum likelihood. The probit model uses the standard normal cumulative distribution function to confine the estimated response probabilities between one and zero (Wooldridge, 2000). We begin our paper with an overview of the existing literature and the history of the use of the yield curve and yield spread as forecasting tools. We will follow this with a discussion of our methods and results before concluding the paper. Hist or y of t he Y ield Sp r ead as a For ecast ing T ool Many studies have shown the yield spread can be useful as a recession forecasting tool. Kessel (1965) was the first to identify the behavior of the term spreads. He studied various spreads between long and short term interest rate yields and found evidence that the spreads became low preceding recessions. Another paper by Butler (1978) also studied the relationship between yields of interest rates, and his results led him to the correct conclusion that no recession would occur in 1979. Both Fama and Bliss (1987) and Mishkin (1990) also provide evidence that a downward sloping yield curve at longer maturities reflects expectations of a recession or a decrease in economic activity. Later studies have examined the ability of the yield curve to predict narrower and alternative measures of economic activity. Estrella and Hardouvelis (1991), for example, suggest that the yield curve can predict gross national product (GNP), consumption, and investment. They compare the forecasting ability of the yield spread over time and find that accuracy improves over longer forecasting horizons—the results are most
significant for horizons of four to six quarters in the future. Chen (1991) also finds that the yield spread can forecast changes in future growth rates of GNP and consumption. He found that the term spread had forecasting power for the growth rate of GNP up to five quarters in the future. Wright (2006) finds similar results when excluding data prior to 1964. He excludes these years because bond rate prices were distorted by being callable. Ang, Piazzesi, and Wei (2006) and Estrella and Trubin (2006) concur with these findings. Many researchers have also tried to pinpoint the exact cause of the yield curve’s strength for predicting recessions. The most apparent source of the predictive power is its link with monetary policy. Estrella and Hardouvelis (1991), Estrella (2004), Estrella and Trubin (2006), and Lahiri and Wang (2006) found that current monetary policy affects the yield curve. When the money supply is tightened or contractionary monetary policy is utilized, short-term interest rates have a tendency to rise. Long-term interest rates, following long-term expectations, also increase, but to a lesser degree than the short-term interest rates. This makes the yield curve flatten or even become inverted. Harvey (1989) hypothesized that expectations of inflation and the resulting Fisher effect is a key determinant of the yield spread. In addition to monetary policy and expectations of inflation, both Wright (2006) and Estrella and Trubin (2006) suggest that taxes, consumer confidence, the stock market, firm confidence, government spending, the number of investors, and risk can influence the demand for assets and, subsequently, the shape of the yield curve. Several papers have also compared the accuracy of the yield curve’s recession forecasts to other well-known forecasting variables. Harvey (1989) studied the slope of yield curve and found that it is a good predictor of real consumption, which affects real economic activity. His results indicate the yield curve inverting in 1969, 1973, 1979, and 1981 before recessions, which followed a few quarters later. He finds that forecasts based on the yield spread are comparable with forecasts from leading econometric models and outperform models derived
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from stock market data. Estrella and Mishkin (1996) compared the yield curve to other leading indicators at different horizons. When compared to the New York Stock Exchange stock price index, the Commerce Department’s Index of Leading Economic Indicators, and the Stock and Watson (1989) index, the yield spread outperforms the preceding models when predicting recessions two or more quarters in the future. In a more recent study, Estrella and Trubin (2006) confirm the slope of the yield curve has accurately predicted each recession between early 1968 and the summer of 2006. The yield spread’s dominance of forecasting can be seen in Figure 1, which depicts the yield spread over 56 years denoting recessions with vertical bars. This graph reinforces the yield curve’s negative slope signals an upcoming recession. During the 1990-91 recession, the yield curve did not perform as well as it did in previous years. The yield spread slope did turn downward but never turned negative. It did not spike as much as it did for recessions before. Researchers can pinpoint two reasons why the yield spread did not meet expectations in 1990-91. The belief is that the 199091 recession was not caused by the same factors as previous recessions. In years prior, the recessions were caused by restrictive monetary policy, which would cause the yield curve to tighten, and,
subsequently, result in a yield spread that had a higher magnification than recessions that were not caused by a restrictive monetary policy. Estrella and Mishkin (1996) explain the other reason: “the amount of variation in the yield curve spread had changed over time and was much less in the 1990s than in the early 1980s, making a strong signal for the 1990-91 recession difficult to obtain.” (p. 4). Rudebusch and Williams (2008) continued to investigate the strength of yield spread to predict recessions. They focused on comparing the yield spread probit model to the SPF from the fourth quarter of 1968 until the first quarter of 2007. Rudebusch and Williams (2008) found that the SPF had around the same predictive power as the yield spread, which is based on real-time estimates, in the current quarter and one quarter in the future. However, when looking more than one quarter into the future and beyond, the yield spread becomes the more accurate predictor of recessions. In summary, the existing literature consistently finds that the yield curve is an accurate predictor of recessions for periods less than 6 quarters into the future. In our paper, we extend the literature by investigating the predictive power of the yield curve 6-quarters and 8-quarters into the future. In the section that follows, we discuss the motivation for our model.
Figur e 1: Y ield -Sp r ead and R ecession Plot
M od el and R esult s As mentioned in previous sections, the variable of choice for measuring the relationship between the yield curve and recessions is the “yield spread” or the difference of interest rates from 10-year US Treasury notes and 3-month US Treasury bills. Estrella and Trubin (2006) and Estrella and Mishkin (1996) state that it is better to use Treasury rates because they are not subject to credit risk premiums. Data for Treasury rates are also readily available. They recommend the 3-month Treasury bill and the 10-year Treasury note as inputs for the probit model because they provide a reasonable combination of robustness and accuracy in predicting recessions. Following the existing literature, we estimate the following simple equation: We estimate the above equation using data beginning with the second quarter of 1953 and ending in the third quarter of 2009. Recession is a binary variable equal to 1 if the United States was in recessing during the quarter, zero otherwise. In econometrics, is referred to as a constant term. There is always at least some probability of a recession. Our forecasted probability is adjusted to this baseline by the inclusion of the constant term. The symbol specifically measures the incremental impact of a one-percentage point change in the yield spread on the probability of a recession, and ε is known as the error term or residual. The term is included in the equation above to note that our equation will not capture all of the variation in the probability of a recession over time. Some of the variation will remain unexplained.
We define the yield spread as explained previously: the difference between the yield on a 10-year Treasury note and 3-month Treasury bill. Since yield data are monthly, we compute an average yield spread for the quarter. We match the yield spread for a specific quarter with the real GDP from 6- and 8-quarters in the future. All GDP figures are in chain-weighted 2005 dollars. We also follow the “R1” and “R2” definitions of recessions described in Rudebusch and Williams (2008). Simply put, recessionary quarters are quarters where real GDP growth is negative according to the “R1” rule. The “R2” rule, on the other hand, follows National Bureau of Economic Research (NBER) definitions more closely and only labels quarters as recessionary if they are negative growth periods for GDP, and they fall precisely after a negative growth quarter growth. In other words, the “R2” rule ignores the first quarter of negative growth in a pairing of two or more consecutive quarters of negative growth. The preceding literature has established the ability of the yield curve to predict recessions less than 6-quarters in the future. We focus on longerterm predictions of 6- and 8-quarters. In addition to forecasting recessions using the “R1” and “R2” rules, we also include estimations using OLS regression and continuous real GDP growth as our dependent variable. Figure 2 displays the results of our estimations. All of our binary coefficients have the expected sign and are statistically significant. The yield curve does not seem to perform as well when modeling recessions using a continuous GDP variable. This result is consistent with the findings of Chauvet and Potter (2005).
Figur e 2: Est im at ion R esult s “R 1” R ule
“R 2” R ule
R eal G DP G r ow t h
0.168
Sp r ead C oefficient 0.425*
0.222
0.164
0.155
0.221
Sp r ead C oefficient
SE
Sp r ead C oefficient
SE
6 qtrs
-0.399***
0.106
-0.462***
8 qtrs
-0.272***
0.0993
-0.435***
*,**, and *** denote statistical significance at the 1%, 5%, and 10% confidence level respectively. SE is an abbreviation for standard error of the coefficient estimate.
SE
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Figur e 3: 6-q uar t er “R 1” R ecession For ecast s
Figur e 4: 6-q uar t er “R 2” R ecession For ecast s
Figur e 5: 8-q uar t er “R 1” R ecession For ecast s
Figur e 6: 8-q uar t er “R 2” R ecession For ecast s
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Using our estimated coefficients from our binary models, we compute predicted recession probabilities and plot them over time. Recessions as defined by the “R1” and “R2” rules are marked in the graphs with vertical gray bars. These graphs are displayed in Figures 3—6. A cursory glance would certainly suggest that the forecasting power is reduced over longer periods of time. Nevertheless, predicted recession probabilities do appear to “spike” at or near recessions on a fairly consistent basis. The data also suggest a few false signals— particularly when following recession rule “R2”. One such example is 2001 when first and third quarter GDP growth was negative, but there were no consecutive quarters of negative GDP growth. Although the recession probability substantially increases using 6- and 8-quarter forecasts, recession rule “R2” does not indicate that a recession occurred during this time period. To further investigate the accuracy of our predictions, we follow techniques outlined by Rudebusch and Williams (2008). We employ the same set of three measures of forecast accuracy: mean absolute error (MAE), root mean square error (RMSE), and the log probability score (LPS). The measures are defined as follows (where PR is the predicted probability of a recession using the yield spread and AR is the actual recession indicator value):
However, our estimations are certainly of comparable magnitude. For example, for a 4-quarter “R1” yield spread forecast Rudebusch and Williams calculate a MAE of 0.209 compared with our estimate of 0.231 and 0.242 for 6- and 8-month forecasts. In addition, our 6- and 8-month forecast errors are smaller than 3-quarter forecasts (0.254) using data from the SPF. Although some degree of accuracy is lost in longer-term forecasting, the yield spread outperforms other forecasts over shorter time periods. Figur e 7: For ecast Er r or C alculat ions 6 qtrs “R 1”
6 qtrs “R 2”
8 qtrs “R 1”
8 qtrs “R 2”
MAE
0.231
0.0831
0.242
0.0839
R M SE
0.340
0.205
0.349
0.206
L PS
0.378
0.163
0.399
0.166
Figur e 8: Est im at es for 2-4 Q uar t er s fr om R ud eb usch and W illiam s (2008)
SPF-M A E SPF-R M SE SPF-L PS
2-q t r s "R 1" 0.248 0.313 0.334
3-q t r s "R 1" 0.248 0.33 0.372
4-q t r s "R 1" 0.258 0.342 0.4
Y S-M A E Y S-R M SE Y S-L PS
0.195 0.303 0.332
0.201 0.307 0.319
0.206 0.311 0.321
SPF and Y S signify Sur v ey of Pr ofessional For ecast er s and y ield sp r ead r esp ect iv ely .
We estimate each measure for our 6- and 8-quarter forecasts following each recession rule. Figure 7 contains the results of this estimation. For shorter time horizons, our calculated forecast errors are larger than those found in Rudebusch and Williams (2008), whose accuracy forecasts can be seen in Figure 8.
C onclusion In this paper, we have examined the accuracy of long term recession forecasts using the yield spread. Although the accuracy of forecasts declines, reductions in accuracy are not substantial when forecasting over 6- and 8-quarter horizons. In fact, the yield curve 6- and 8-quarter forecasts outperform forecasts made by the SPF over 4quarter horizons or shorter. Investors and policy makers alike stand to gain insight into the future
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state of the economy by paying attention to the yield curve. Recessions can be predicted with a fair amount of accuracy as far as two years into the future using the yield curve. Not es 1. All GDP data was taken from The Bureau of Economic Analysis “Percent Change from Previous Period” using GDP Percent Change based on chained 2005 dollars. 2. All yield curve data was taken from http://finance.yahoo.com using Secondary Market Treasury Bonds at maturities of 3-month and 10 years from the second quarter of 1953 to the third quarter of 2009. W or k s C it ed
Ang, A., Piazzesi, M., & Wei M. (2006). What does the yield curve tell us about GDP growth? Journal of Econometrics, 131, 359-403. Butler, L. (1978). Recession? - A market view. Federal Reserve Bank of San Francisco Weekly Letter, December 15, 1-3. Chauvet, M., Potter, S. (2005). Forecasting recessions using the yield curve. Journal of Forecasting, 24(2), 77-103. Chen, N. (1991). Financial investment opportunities and the macroeconomy. The Journal of Finance, 46(2), 529-554. Estrella, A. (2004). Why does the yield curve predict output and inflation? The Economic Journal, 115, 722-744. Estrella, A. (2005). The yield curve as a leading indicator: frequently asked questions. Federal Reserve Bank of New York, October 2005, 1-23. Estrella, A., Hardouvelis, G. A. (1991). The term structure as a predictor of real economic activity. Journal of Finance, 46(2), 555-576.
Estrella, A., Mishkin, F. S. (1996). The yield curve as a predictor of U.S. recessions. Current Issues in Economics and Finance, 2(7), 1-6. Estrella, A., Mishkin, F. S. (February 1998). Predicting U.S. recessions: Financial variables as leading indicators. The Review of Economics and Statistics, 80(1), 45-61. Estrella, A., Trubin, M. R. (2006). The yield curve as a leading indicator: Some practical issues. Current Issues in Economics and Finance, 12(5), 1-7. Fama, E. F., Bliss, R. R. (1987). The information in longmaturity forward rates. The American Economic Review, 77(4), 680-692. Fernald, J., Trehan, B. (2006). Is a recession imminent? Federal Reserve Board of San Francisco Economic Letter, 2006 (32), 1-3. Harvey, C. R. (Sep./Oct. 1989). Forecasts of economic growth from the bond and stock markets. Financial Analysts Journal, 45(5), 38-45. Kessel, R. A., (1965) The cyclical behavior of the term structure of interest rates. National Bureau of Economic Research Occasional Paper No. 91. Lahiri, K., Wang, J. G. (2006). Subjective probability forecasts for recessions. Business Economics, 41(2), 2637. Mishkin, F. S. (1990). The information in the longer maturity term structure about future inflation. The Quarterly Journal of Economics, 105(3), 815-828. Rudebusch, G. D., Williams, J. C. (July 2008). Forecasting recessions: The puzzle of the enduring power of the yield curve. Journal of Business and Economic Statistics, 27(4), 492-503. Stock, J., Watson, M. (1989). New indexes of coincident and leading indicators in O. Blanchard and S. Fischer, eds., NBER Macroeconomic Annual 4. Wooldridge, J. M. (2000). Introductory economics: A Modern Approach. United States of America: Thomson Learning. Wright, J. H. (2006). The yield curve and predicting recessions. Washington, D.C.: Divisions of Research & Statistics and Monetary Affairs, Federal Reserve Board, Finance and Economics Discussion Series Paper.
Jennifer Sab ol ('11) is an Accounting major at Saint Francis. She has been named to the Dean’s List for all semesters and is a member of Sigma Delta Beta, Kappa Mu Epsilon National Honor Society for Mathematic Excellence, and Delta Epsilon Sigma. After graduation, she plans to attend graduate school to earn a Master’s of Accountancy. Her career goal is a position in academia or business. This is her first collaborative academic paper, and she is currently working on a second paper.
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A n Inv ar iant of M et r ic Sp aces Und er Bor nologous Eq uiv alences Brittany N. Miller Department of Mathematics, Computer Science, and Engineering School of Sciences bnmst6@mail.francis.edu
Laura M. Stibich Department of Mathematics, Computer Science, and Engineering School of Sciences lmsst8@mail.francis.edu
Julia H. Moore Department of Mathematics, Computer Science, and Engineering School of Sciences jhmst5@mail.francis.edu
Brendon L. LaBuz, Ph.D. Department of Mathematics, Computer Science, and Engineering School of Sciences bll001@mail.francis.edu
Bornologous equivalence is defined as two metric spaces that are bornologous functions that also map onto one another. This equivalence is related to but also a more restrictive concept than the mathematical concept of a coarse equivalence. We give examples of spaces that are bornologously equivalent. To show that two spaces are bornologously equivalent, it is necessary to find a function from one to the other that serves as an equivalence. By definition, two spaces are not bornologously equivalent if there is no function from one to the other that serves as an equivalent. To allow us to show that such a function does not exist, we define an invariant that detects the number of ways that sequences can go to infinity in a space. This invariant is only for a certain class of spaces that we call sigma stable. Since this number is invariant under bornologous equivalences, it allows for us to determine that two spaces are not bornologously equivalent. Keywords: Metric space, Bornologous, Coarse equivalence Int r od uct ion A function is bornologous if for every there is an such that if , then , where N and M are intervals between points in X and Y, respectively. Two metric spaces X and Y are bornologously equivalent if there are two bornologous functions and such that and . Determining whether two metric spaces are bornologously equivalent uses large scale geometry, unlike calculus which focuses on small scale geometry. When using small scale geometry, the step function would not be considered a continuous function. Large scale geometry; on the other hand, does show the step function to be continuous. Since bornologous equivalence is related to but also a more restrictive
concept than coarse equivalence, if two metric spaces are bornologously equivalent, then they are also coarsely equivalent. R esult s An example that illustrates that bornologous equivalence is a large scale property is the graph of the step function where is the floor of . We show that is bornologoulsy equivalent to the real numbers . We consider the natural function that sends to (Figure 1). This function is bornologous as is its inverse, the function that sends to . Since these two spaces are bornologously equivalent they have the same large scale structure.
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Figur e 1. T he gr ap h of t he st ep funct ion is b or nologoulsy eq uiv alent t o t he r eal num b er line.
Now let the metric space be the union of the sets , , and . We call the vase, which is an example of a space that should not be bornologously equivalent to R . We consider a natural function defined as
(Figure 2). This function flattens (or the vase) out onto the real line. Notice that two points that are on opposite sides of the vase get sent by f to points that are far apart on R , making the function not bornologous. In fact, the farther up we are on V, the farther apart the image points become on R . Because this function is not bornologous, we can also show that the two metric spaces are not bornologously equivalent. In order to show that two metric spaces are not bornologously equivalent, we need to show that every function from the two metric spaces is not bornologous. This task is not feasible because there is no way to test every function.
Figur e 2. T he v ase should not b e b or nologously eq uiv alent t o t he r eal num b er s.
In order to show that two spaces are not bornologously equivalent, we needed to determine a way to detect the equivalence. The method we use to determine that metric spaces are not bornologously equivalent is to detect the number of ways the metric space goes to infinity. To do so we used sequences in a metric space. Given , an -sequence in a metric space is a list of elements of X such that for all . We say the sequence is based at , and we call the base point of . To help detect the number of ways of going to infinity, we will use sequences that are infinite. An N-sequence goes to infinity, , if . This definition means that the distance from the base point tends to go to infinity. The set of all sequences in based at that go to infinity is . W e us e as our motivation for defining what it means for two sequences to be equivalent. We want any two sequences in whose limits are infinity to be equivalent. Similarly, we want any two sequences whose limits are negative infinity to be equivalent. However, we do not want a sequence whose limit is infinity to be equivalent to a sequence whose limit is negative infinity.
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Figur e 3. T w o noneq uiv alent seq uences in R
We define two sequences to be equivalent as follows. Define and to b e related, , if is a subsequence of or is a subsequence of . If is a subsequence of , we say that is a supersequence of . Define and to be equivalent, , if there is a finite list of elements such that . This definition means that can be transformed into b y us i n g subsequences and supersequences. Let denote the equivalence class of in and be the set of equivalence classes. We are interested in counting the number of different types of sequences that go to infinity, i.e., the cardinality (size) of . Now if , an -sequence is an -sequence so we can define a function . If these functions are eventually one-to-one correspondences we define the space as -stable. A space being stable essentially means that has the same cardinality for large , which is the invariant we considering. We denote this cardinality to be . Suppose is bornologous between metric spaces and is a basepoint of and . Also, suppose and are -stable. Since they are -stable, . We also prove that if two points can be joined by an -chain (the finite version of the -
sequence), then . Therefore, if two -stable spaces have different values, they cannot be bornologously equivalent. We show that is -stable and that so that . We show that is -stable and that Since , bornologously equivalent.
so that and
are
. not
Figur e 4. T w o eq uiv alent seq uences in V
Discussion Resulting from our research, we have determined that the vase and the real line are not bornologously equivalent. Because bornologous equivalence and coarse equivalence are related, does our theory for the vase and the real line create an invariant under coarse equivalence? For further study, we, also, suggest trying to calculate and to find an example of a metric space that is not sigma stable. R efer ences
Roe, John. Lectures on coarse geometry, University lecture series, 31. American Mathematical Society, Providence, RI, 2003.
Br it t any N. M iller (’12) is a Mathematics/Secondary Education major. She hopes to be a high school or middle school Mathematics teacher in the area. She enjoys reading, watching movies, and having fun. Julia H. M oor e (’12) is a Mathematics major with a Physics minor. She plans to attend graduate school after graduation and become a meteorologist. Her interests are field hockey, running, and reading. L aur a M . St ib ich (’12) is a Mathematics Secondary Education major. She hopes to teach Mathematics in a secondary school, hopefully early high school. Her interests include sewing and watching the Disney Channel
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R unning int o t he Unk now n: T he R eligious Dim ensions of Dist ance R unning Andrew R. McKee Religious Studies Department School of Arts and Letters armst8@mail.francis.edu
Arthur Remillard, Ph.D. Religious Studies Department School of Arts and Letters aremillard@francis.edu
“There's something so universal about that sensation, the way running unites our two most primal impulses: fear and pleasure. We run when we're scared, we run when we're ecstatic, we run away from our problems and run around for a good time. And when things look worst, we run the most.” –Christopher McDougall, Born to Run1 Religious1theorist Joseph Campbell began his running career at Columbia University while working on his undergraduate studies. During this period of his life, running became more than just a physical activity for Campbell. It helped him form close bonds with other runners, and allowed him to achieve a heightened state of being. In interviews compiled in The Hero’s Journey, he recalled his track days discussing the relationship he felt between others and the heightened awareness running provided him. He stated that, “As I look back now there were a couple moments in the last eighty yards of the half mile when I was running in championship time…you know you’re spaced out by then… If anyone would ask me what the peak experiences in my life were…those races would be it; more than anything in my whole life.”2 Campbell is far from alone in the running community. This paper examines the feeling of exhilaration achieved while running. The feeling plays a central role in expressing how running can take on more meaning and become more than simple physical exertion. Many runners feel a reconnection with nature, themselves, and with others after spending long periods while performing the simple act of running. This
experience can take on distinctly religious qualities. In each case, elements of scene, setting, disposition and work are all present. For running to take on religious qualities, one must be open to the experience of running. Much like a church experience, the outcome of the event is dependent on the person. For a higher state of awareness to occur in distance running, there also must exist an everyday plane to separate from. Mircea Eliade’s ideas of sacred and profane are at the forefront of the distinction between an everyday run, and the runner’s high or religious experience. The profane makes up the realm of everyday business of things ordinary, random and largely unimportant, the mundane; on the other hand, the sacred is the sphere of supernatural, things extraordinary, memorable, and momentous.3 Running and the major sports at large in America, as demonstrated by Michael Novak, Alan Guttman, and Joseph L. Price, seems to promote contact with the sacred.4 For Eliade, athletic events, great public spectacles of our time, show similar affinities when compared to the ceremonies of archaic peoples.5 They generate intense emotions and center on a single 3
1
McDougall, Christopher, Born to Run: A Hidden Tribe, Super Athletes and the Greatest Race the World has Never Seen (New York: Alfred A. Knopf, 2009), 11. 2 Brown L, Stuart, and Phil Cousineau, The Hero’s Journey: The World of Joseph Campbell (New York: Harper & Row, 1990), p. 23.
Pals, Daniel, 8 Theories of Religion (Oxford University Press, USA: January 12, 2006), p. 199. 4 Here, Novak’s Joy of Sports, Guttman’s From Ritual to Record: the Nature of Modern Sports and Price’s Rounding the Bases all provide in depth examinations of the holy trinity sports – baseball, football, and basketball - as civil religions. 5 Pals, 219.
17 “concentrated” game time, very much like the sacred moments of primitive ritual.6 One way to signify this distinction is using Ninian Smart’s discussion of the experiential dimension of religion. According to Ninian Smart in his book Worldviews: Cross Cultural Exploration of Human Beliefs, religions contain six distinct dimensions, and while they must not be present in each religious practice, they provide the guidelines for a religion in action.7 The major religious aspect examined in relation to long distance running is the experiential. According to Smart, this dimension is not confined to a place of worship and it is possible to, “get a sense of the numinous God outside the cathedral, church, or temple – in nature. This element is important because it signifies the difference between a religion and a religious experience. A religious experience does not require a traditional holy setting, and its more dependent on the individual involved. Religious thinkers and believers have long heard the ‘voice of God’ in the wind on the tips of the soaring mountains, or in the churning of the ocean.”8 Distance runners may not belong to any religion or worship a divine being; however, they are not exempt from an overpowering feeling of other worldliness. Runners who disassociate themselves from other concerns allowing for a reconnection with themselves and nature to experience an elevated state of awareness share in the experiential dimension. One part time runner, living in Washington D.C, used running to depart from the city, reinvigorating his spirits and causing him to feel, “a perfect peace of being, of living in the moment, unencumbered and liberated.”9 Running alone allowed this runner to connect to something inside himself, which he could not find in everyday life. Another athlete concluded that, “it’s easier to 6
Ibid. Doctrinal, Mythological, Ethical, Ritual, Experiential, Institutional, and Experiential 8 Smart, Ninian. Worldviews: Cross-cultural Explorations of Human Beliefs (New Jersey: Prentice Hall, 1995), 59. 9 Handan Tulay Satiroglu, “Metamorphous,” in The Runner’s High: Illumination and Ecstasy, ed. Garth Battista (New York: Breakaway Books, 2004), 25. 7
reach such highs if you unplug from your iPod and connect to nature, which also means I’ll probably never find highs on a treadmill.”10 For Smart, these runners’ peace and being is similar to what occurs in a mystical experience. In such, there exists a pure consciousness; there is not much importance to God or the gods but the highest value on personal liberation.11Scene, disposition, setting and working hard all came together causing the runner to have a liberating sensation. According to doctor, runner, and philosopher George Sheehan many people think of religion as something coming from the past affecting ourselves in the future. He feels that we ignore what is important: the present.12In his mind, by focusing on what is important in the present one can, “reconcile the separations of body, of pain and pleasure, of the conscious and the unconscious… a way to make the ordinary extraordinary; the commonplace, unique; the everyday, eternal.”13 He described his shedding of everyday norms by writing that, “running made me free. It rid me of concern for the opinion of others. Dispensed me from rules and regulations imposed from outside. Running let me start from scratch.”14 For Sheehan, running each day allowed for a fresh start on life. This new beginning each day permitted every run to be something new and exciting, for him, a truly religious expression. By becoming utterly present, Sheehan may have expressed what Max Weber called an “ecstatic state.” This special experience puts people beyond the realm of everyday activity and discloses to them another level of reality.15 For Sheehan and many other runners, running becomes
10
Larry Erlich, comment on ““Yes, Running Can Make You High,” comment posted March 27, 2008, http://community.nytimes.com/comments/www.nytimes.com/ 2008/03/27/health/nutrition/27best.html?sort=oldest&offset= 1 11 Smart, 61. 12 Sheehan, George, Running and Being: The Total Experience (New York: Simon and Schuster, 1978), 112. 13 Ibid 14 Ibid, 25. 15 Pals, Daniel, Eight Theories of Religion (New York: Oxford University Press, 2006), 166.
18 the total experience, being present, of forgetting about past and future. “12 Miles is Always 12 Miles” –Quenton Cassidy, Once a Runner16 The fictional running heroes found in novels throughout the running community echo real world runners statements on mystical experiences. In the novel Once a Runner, the standout runner Quenton Cassidy felt that the last one hundred meters of the mile race, “make him wonder if his life could ever be quite the same again as it was now, while he was so vital, so quick, so near immortal.”17 A mystical feeling found while running on the track gave Cassidy the chance to become like the heroes he idolized in the past. In his race, Cassidy broke the mythical four-minute mile barrier while in the process beating an Olympic gold medalist. Here, his mental state paralleled Joseph Campbell’s describing a state of being where they both flew along with no conscious effort.18 This realization of a heighted state is imperative to a mystic experience.19 For many runners the heightened awareness achieved is developed in the solitude of running alone. The phrase, “The Loneliness of the Long Distance Runner” made famous by the 1959 Alan Sillitoe novel of the same name, highlights this alertness. In the story, the protagonist Colin, is arrested for robbery and sent to a reformatory school. While locked up, he starts to seek solace in distance running. Each day Colin ran alone, running over creeks and through the woods for no other reason than to recapture the feeling of freedom he once possessed. In doing so he stated that, “I knew what the loneliness of the long distance runner running across country felt like, realizing that as far as I was concerned this feeling was the only honesty and realness there was in the
16
John L. Parker, “Once a Runner,” in The Runner’s Literary Companion: Great Stories and Poems about Running, ed. Garth Battista (New York City: Breakaway Books, 1994), 12. 17 Ibid, 8. 18 Ibid, 9. 19 Smart, 62.
world.”20 Running permitted Colin to achieve brief moments of liberty from his incarcerators and gave him the chance to reflect on his status. Like many other runners, this state allowed for a period of reflection and peace. While not a directly religious action, running alone allows this runner to become in touch with himself in a spiritual way in a way that went beyond simple physical action. In both cases of runners telling their personal stories and of stories found in running lore, Ninian Smart’s discussion of mystical experience and the experiential dimension is present. In each case, runner’s have found that being alone in nature or at least away from daily concerns affords them the best chance to experience running. For cynics of the religious experience, Smart raises another consideration: How do we tell what belongs to the experience itself, and what to the interpretation?21 He evokes an example of imagery, “If I see a rope on the ground and perceive it as a snake…isn’t it true to say I experienced a snake? So, if a mystic sees the inner light of consciousness…does he not then experience God?22 He asserts that it may be so, a single type of mystical experience allows one to understand that there are reoccurring patters of inner consciousness into which, depending on context, different cultures read different meaning.23 For a non – runner, distance training possessing a religious quality may seem absurd; however, to one who has achieved an altered state, the experience is very real and exhilarating. "My Husband used to be a Methodist. Now he's a runner."24 Distance running, with its focus on the individual, can still fall under the same distinction as the major team sports. In both cases, it is 20
Alan Sillitoe, “The Loneliness of the Long-Distance Runner,” in The Runner’s Literary Companion: Great Stories and Poems about Running, ed. Garth Battista (New York City: Breakaway Books, 1994), 189. 21 Smart, 65. 22 Ibid. 23 Ibid. 24 Marc Bloom, “Spiritual Movement: Running isn’t a Religion but It Can Feel Like One,” Runner’s World, September 10, 2006.
19 possible to achieve a religious break in a normal routine. Each day the rituals of running allow for an increased encounter with the sacred, a chance for a break from the profane. After reaching such a peak one female runner wrote that, “I experienced a fantastic, expanding sense of joy. I could not restrain myself from reaching my arms up, palms to the sun, to celebrate the sheer pleasure of being alive and propelling myself forward through the humid air.”25 While not occurring on each run, this woman found that from time to time, running brought her to a higher state, taking on a more significant meaning be it a connection to some outside force or achieving a new level of being. While focusing on Ninian Smart’s ideal of mystical experience and the experiential dimension, a broader discussion of distance running in the religions realm is still necessary. However, within the running community, from weekend warriors to the professional levels, there exists a connection on the experiential dimension of religion. Christopher McDougall writes about this connection in his book Born to Run. Here, the author himself treks to the mountains of a remote Mexican Village and immerses himself into the culture of the Tarahumara running tribe. After several years of learning the culture and norms of the society, McDougall sets up a race between several members of the running tribe and a handful of the United State’s top ultra marathon runners.26 However, while the competition was intense and neither side wished to cede victory, the participants each found connections with each other to be more important than the eventual outcome. McDougall found that during races, “the reason we race isn't so much to beat each other, he understood, but to be with each other.”27 He found peace in racing and pushing himself alongside, not against, other runners. This cross-cultural connection allows for a common plane on which to experience what Campbell thought about when he discussed the mystical bliss
that occurs when the body is overtaxed.28 Again, the runner’s high or religious connections among people spanned the running community. This is by no means an all-encompassing means of studying the phenomena of the runner’s high; however, the religious elements found while experiencing this feeling provides a unique window into the study of religion outside of traditional settings. The runner’s high spans the entire running community from weekend warrior runners to elite professionals. Perhaps where a study such as this is most valuable is showing the connection between sports and religious experience, and expanding on the study the spiritual lives of American religion. The best way to expand this study is for one to take part in it themselves, and I encourage everyone to strap on a pair of running shoes, head for trails, and get running. A nd r ew M cK ee ('11) is a History and Political Science major. He plans to attend graduate school next fall. He is a member of the Saint Francis Cross Country, Indoor Track and Outdoor track teams.
25
Bloom. Note: an Ultra Marathon is typically a 50 kilometer to 100 mile race, and is usually contested over some inhospitable landscape. 27 McDougall, 253. 26
28
Brown L, Stuart, and Phil Cousineau, 22.
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