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Augustan Era Policy on the Rhine Frontier from 34 B.C.E.-16 C.E. Kyle McStay References Primary Sources 1. Cassius Dio. The Roman History: Books 48-49: Trans. Earnest Cary for Loeb Classical Library, 1914-1927. Text on LacusCurtius at http://penelope.uchicago.edu/Thayer/E/Roman/Texts/ Cassius_Dio/home.html Books 50-56: Trans. Ian Scott-Kilvert. London: Penguin Books, 1987. 2. Gaius Velleius Paterculus. The Roman History. Trans. Fredrick W. Shipley for Loeb Classical Library, 1924. Text on LacusCurtius at http://penelope.uchicago.edu /Thayer/E/Roman/Texts/Velleius_Paterculus/home.html 3. Publius Cornelius Tacitus. Annals. Trans. Alfred John Church and William Jackson Brodribb, 1942. E-text edition by Bruce. J. Butterfield, 1997. http://mcadams. posc.mu.edu/txt/ah/tacitus/ Secondary Sources 1. Gruen, Erich S. “The Expansion of the Empire Under Augustus.” The Cambridge Ancient History, 2nd ed. Vol. 10. Eds. Alan K. Bowman, Edward Champlin, and Andrew Lintott. Cambridge, New York, Melbourne: Cambridge University Press, 1996. pp. 147-197. 2. Ruger, C. “Germany.” The Cambridge Ancient History, 2nd ed. Vol. 10. Eds. Alan K. Bowman, Edward Champlin, and Andrew Lintott. Cambridge, New York, Melbourne: Cambridge University Press, 1996. pp. 147-197. pp. 517-534. 3. Wells, C[olin] M[ichael]. The German Policy of Augustus. Oxford: Clarendon Press, 1972. Endnotes 1 For detailed discussion of this view see: Luttwak, Edward N. The Grand Strategy of the Roman Empire, From the First Century A.D. to the Third. Baltimore and London: Johns Hopkins University Press, 1976. 2 For detailed criticism of the views of Luttwak see: Mattern, Susan P. Rome and the Enemy: Imperial Strategy in the Principate. Berkeley, Los Angeles, London: University of California Press, 1999. 3 Gruen, pp.169-171, 178-188 4 The area known to the Romans as “Gaul” roughly encompassed the territory of modern-day France and Belgium. 5 Dio LI.21.5-6 6 Dio XLVIII.49.3 7 Dio LIV.11.2 8 Dio XLVIII.49.3, LI.21.5-6 9 Velleius Paterculus II.97.1 10 Dio LIV.20.4 11 Wells, pp. 93-148 12 Wells, pp. 149-154 13 Haltern is likely, but not certainly, the fortress referred to as “Aliso” by Cassius Dio, Velleius Paterculus, and Tacitus (see Wells, pp. 152-153, pp.192-198, for discussion of available evidence). 14 Wells, pp. 99-100 15 Gruen, p. 179 16 Wells, pp. 94-95 17 Dio LIV.32.1 18 Dio LIV.32.1-3 19 Dio LIV.33.1-5; Gruen p. 181 20 Gruen, p. 181; Dio LIV.36.4 21 Dio LIV. 36.4 22 Dio LV.1.2-5; Velleius Paterculus II.97.2-3 23 Velleius Paterculus II.97.4 24 Tacitus, Annals, 1.57.2 25 Dio LVI.18.2 26 Wells, pp. 238-239 27 Wells, p. 239 28 Wells, p. 239 29 Dio LVI.18.3; Velleius Paterculus II.118.1 30 Velleius Paterculus II.118.1 31 Dio LVI.18.5 32 Ruger, p. 527 33 Dio LVI.20.1-3; Velleius Paterculus II.119 34 Dio LVI.19.4-20.5; Velleius Paterculus II.119 35 Velleius Paterculus II.120.4 36 Velleius Paterculus II.120.3; Dio LVI.22.2 37 Tacitus, Annals 1.3.6 38 Wells, pp. 198-206 for detailed discussion of available evidence. 39 Wells, p. 244-245 40 Wells, p. 243-244 41 Dio LVI.33.5 34 UCR Un d e r g r a d u a t e Re s e a r c h Jo u r n a l
Fractal Strings and Number Theory: The Harmonic String and the Prime String Jason C. Payne, Michel L. Lapidus Department of Mathematics University of California, Riverside Abstract The purpose of this paper follows two veins which converge at one critical juncture- a bridge between two seemingly disparate concepts. The first goal is to provide a brief survey of the topics of fractal strings and their complex dimensions, which is achieved through the introduction of their geometric and spectral zeta functions. Following this is consideration of two important examples of generalized fractal strings- the harmonic string and the prime string. Through this two discussions, the goal is to establish a strong connection between the concrete study of the geometry and spectrum of fractal strings and the abstract world of number theory, which is achieved by way of the infamous Riemann zeta function. Faculty Mentor Michel L. Lapidus Department of Mathematics Michel Lapidus is Professor of Mathematics at UCR and also teaches in the departments of Physics, Electrical Engineering and Computer Science. He works at the crossroad of many research areas, including Mathematical Physics, Fractal Geometry, Dynamical Systems, Parital Differential Equations, Noncommutative Geometry, and Number Theory. His recent research books include “The Feynman Integral and Feynman’s Operational Calculus” (Oxford Univ. Press, 2000, paperback: 2001; joint with G. W. Jonson), “Fractal Geometry and Number Theory” (Birkhauser, 2000), “Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and spectra of fractal strings” (Springer-Verlag, 2006) [both joint with M. van Frankenhuysen], and most recently, “In Search of the Riemann Zeros: Strings, fractal membranes and noncommutative spacetimes” (Amer. Math. Soc., Jan. 2008). Professor Lapidus is a Fellow of the American Association for the Advancement of Science, and has been a member of the American Mathematical Society Council since January 2002. Over the last nine years, he has worked with nine undergraduate research projects and undergraduate Honors Theses. A U T H O R Jason C. Payne Pure Mathematics Jason Payne is a graduating senior majoring in Pure Mathematics. His research interests include fractal geometry and complex analysis and their applications in both analytic and algebraic number theory, and Riemannian geometry. He is currently working on a senior thesis on volume formulas in arbitrary (potentially fractal) dimensions, and how they can be combined with the study of fractal strings in order to provide a generalization of Gauss’s Circle Problem. Some fellow math majors and he are creating of an official math club at UCR. After graduating this spring, Jason will begin the Mathematics Ph.D. program here at UCR where he will continue his research in fractal geometry, complex analysis, and number theory. UCR Un d e r g r a d u a t e Re s e a r c h Jo u r n a l 35
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