Skip to main content Accessibility help
×
Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-16T14:20:17.001Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  21 April 2021

Tom Leinster
Affiliation:
University of Edinburgh
Get access
Type
Chapter
Information
Entropy and Diversity
The Axiomatic Approach
, pp. 412 - 430
Publisher: Cambridge University Press
Print publication year: 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aczél, J.. On mean values. Bulletin of the American Mathematical Society, 54(4):392400, 1948.CrossRefGoogle Scholar
[2] Aczél, J.. Lectures on Functional Equations and their Applications. Academic Press, New York, 1966.Google Scholar
[3] Aczél, J. and Daróczy, Z.. On Measures of Information and Their Characterizations, volume 115 of Mathematics in Science and Engineering. Academic Press, New York, 1975.Google Scholar
[4] Adler, R. L., Konheim, A. G., and McAndrew, M. H.. Topological entropy. Transactions of the American Mathematical Society, 114:309319, 1965.Google Scholar
[5] Aitchison, J.. Simplicial inference. In Viana, M. A. G. and Richards, D. S. P., editors, Algebraic Methods in Statistics and Probability, volume 287 of Contemporary Mathematics, pages 122. American Mathematical Society, Providence, RI, 2001.Google Scholar
[6] Alesker, S.. Theory of valuations on manifolds: a survey. Geometric and Functional Analysis, 17:13211341, 2007.Google Scholar
[7] Alesker, S., Artstein-Avidan, S., and Milman, V.. Characterization of the Fourier transform and related topics. Comptes Rendus de l’Académie des Sciences, Paris, Series I, Mathématique, 346:625628, 2008.Google Scholar
[8] Alesker, S. and Fu, J. H. G.. Integral Geometry and Valuations. Advanced Courses in Mathematics CRM Barcelona. Birkhäuser, Basel, 2014.Google Scholar
[9] Alsedà, L., Llibre, J., and Misiurewicz, M.. Combinatorial Dynamics and Entropy in Dimension One, volume 5 of Advanced Series in Nonlinear Dynamics. World Scientific, Singapore, 2nd edition, 2000.Google Scholar
[10] Amari, S.. Differential geometry of curved exponential families – curvatures and information loss. Annals of Statistics, 10(2):357385, 1982.Google Scholar
[11] Amari, S.. A foundation of information geometry. Electronics and Communications in Japan, 66-A(6):110, 1983.CrossRefGoogle Scholar
[12] Amari, S.. Information Geometry and its Applications, volume 194 of Applied Mathematical Sciences. Springer, Tokyo, 2016.Google Scholar
[13] Amari, S. and Nagaoka, H.. Methods of Information Geometry, volume 191 of Translations of Mathematical Monographs. Oxford University Press, Oxford, 1993.Google Scholar
[14] Ancrenaz, M., Gumal, M., Marshall, A. J., Meijaard, E., Wich, S. A., and Husson, S.. Pongo pygmaeus. IUCN Red List of Threatened Species e.T17975A17966347, 2016.Google Scholar
[15] Apostol, T. M.. Mathematical Analysis. Addison-Wesley, Reading, MA, 1957.Google Scholar
[16] Apostol, T. M.. Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics. Springer, New York, 1976.Google Scholar
[17] Arimoto, S.. Information measures and capacity of order α for discrete memoryless channels. In Topics in Information Theory: 2nd Colloquium, Keszthely, Hungary, 1975, pages 41–52. North-Holland, Amsterdam, 1977.Google Scholar
[18] Arnold, V. I.. Mathematical Methods of Classical Mechanics, volume 60 of Graduate Texts in Mathematics. Springer, New York, 2nd edition, 1989.Google Scholar
[19] Asao, Y.. Magnitude homology of geodesic metric spaces with an upper curvature bound. Algebraic & Geometric Topology, to appear, 2021.Google Scholar
[20] Aubrun, G. and Nechita, I.. The multiplicative property characterizes p and Lp norms. Confluentes Mathematici, 3:637647, 2011.Google Scholar
[21] Avery, T.. Structure and Semantics. PhD thesis, University of Edinburgh, 2017.Google Scholar
[22] Ay, N., Jost, J., , H. V., and Schwachhöfer, L.. Information Geometry, volume 64 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Cham, 2017.Google Scholar
[23] Baez, J. and Dolan, J.. From finite sets to Feynman diagrams. In Engquist, B. and Schmid, W., editors, Mathematics Unlimited – 2001 and Beyond, pages 2950. Springer, Berlin, 2001.CrossRefGoogle Scholar
[24] Baez, J. and Fritz, T.. A Bayesian characterization of relative entropy. Theory and Applications of Categories, 29:421456, 2014.Google Scholar
[25] Baez, J., Fritz, T., and Leinster, T.. A characterization of entropy in terms of information loss. Entropy, 13:19451957, 2011.CrossRefGoogle Scholar
[26] Bakker, M. G., Chaparro, J. M., Manter, D. K., and Vivanco, J. M.. Impacts of bulk soil microbial community structure on rhizosphere microbiomes of Zea mays. Plant Soil, 392:115126, 2015.Google Scholar
[27] Banach, S.. Sur l’équation fonctionnelle f (x + y) = f (x) + f (y). Fundamenta Mathematicae, 1:123124, 1920.Google Scholar
[28] Barceló, J. A. and Carbery, A.. On the magnitudes of compact sets in Euclidean spaces. American Journal of Mathematics, 140(2):449494, 2018.CrossRefGoogle Scholar
[29] Barron, A. R.. Entropy and the central limit theorem. Annals of Probability, 14(1):336342, 1986.Google Scholar
[30] Barton, B. H. and Moran, E.. Measuring diversity on the Supreme Court with biodiversity statistics. Journal of Empirical Legal Studies, 10:134, 2013.CrossRefGoogle Scholar
[31] Batanin, M. A.. The Eckmann–Hilton argument and higher operads. Advances in Mathematics, 217:334385, 2008.Google Scholar
[32] Baudot, P. and Bennequin, D.. The homological nature of entropy. Entropy, 17:32533318, 2015.CrossRefGoogle Scholar
[33] Beck, C. and Schlögl, F.. Thermodynamics of Chaotic Systems: An Introduction, volume 4 of Cambridge Nonlinear Science Series. Cambridge University Press, Cambridge, 1993.Google Scholar
[34] Bénabou, J.. Introduction to bicategories. In Bénabou, J., Davis, R., Dold, A., Mac Lane, S., Isbell, J., Oberst, U., and Roos, J.-E., editors, Reports of the Midwest Category Seminar, Lecture Notes in Mathematics 47. Springer, Berlin, 1967.CrossRefGoogle Scholar
[35] Bereziński, P., Jasiul, B., and Szpyrka, M.. An entropy-based network anomaly detection method. Entropy, 17:23672408, 2015.Google Scholar
[36] Berger, C. and Leinster, T.. The Euler characteristic of a category as the sum of a divergent series. Homology, Homotopy and Applications, 10(1):4151, 2008.Google Scholar
[37] Berger, W. H. and Parker, F. L.. Diversity of planktonic Foraminifera in deep-sea sediments. Science, 168:13451347, 1970.Google Scholar
[38] Bhattacharyya, A.. On a measure of divergence between two statistical populations defined by their probability distribution. Bulletin of the Calcutta Mathematical Society, 35(1):99109, 1943.Google Scholar
[39] Bishop, R. L.. Elasticities, cross-elasticities, and market relationships. American Economic Review, 42(5):780803, 1952.Google Scholar
[40] Blackwell, R., Kelly, G. M., and Power, A. J.. Two-dimensional monad theory. Journal of Pure and Applied Algebra, 59:141, 1989.Google Scholar
[41] Blass, A. and Gurevich, Y.. Negative probability. Bulletin of the European Association for Theoretical Computer Science, 115:126142, 2015.Google Scholar
[42] Blass, A. and Gurevich, Y.. Negative probabilities, II: What they are and what they are for. Bulletin of the European Association for Theoretical Computer Science, 125:152168, 2018.Google Scholar
[43] Boardman, J. M. and Vogt, R.. Homotopy Invariant Algebraic Structures on Topological Spaces, volume 347 of Lecture Notes in Mathematics. Springer, Berlin, 1973.Google Scholar
[44] Borceux, F.. Handbook of Categorical Algebra 2: Categories and Structures, volume 51 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1994.Google Scholar
[45] Borges, E. P.. On a q-generalization of circular and hyperbolic functions. Journal of Mathematical Physics A: Mathematical and General, 31:52815288, 1998.Google Scholar
[46] Borgoo, A., Jaque, P., Toro-Labbé, A., Alsenoy, C. V., and Geerlings, P.. Analyzing Kullback–Leibler information profiles: an indication of their chemical relevance. Physical Chemistry Chemical Physics, 11:476482, 2009.Google Scholar
[47] Borwein, J. M. and Lewis, A. S.. Convex Analysis and Nonlinear Optimization: Theory and Examples. Canadian Mathematical Society Books in Mathematics. Springer, New York, 2000.CrossRefGoogle Scholar
[48] Boularias, A., Kober, J., and Peters, J.. Relative entropy inverse reinforcement learning. In Gordon, G., Dunson, D., and Dudík, M., editors, Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, Fort Lauderdale, FL, USA, 11–13 April 2011, volume 15 of Proceedings of Machine Learning Research, pages 182189. PMLR, Fort Lauderdale, FL, 2011.Google Scholar
[49] Box, G. E. P. and Cox, D. R.. An analysis of transformations. Journal of the Royal Statistical Society, Series B (Methodological), 26:211–252, 1964.CrossRefGoogle Scholar
[50] Bromaghin, J. F., Rode, K. D., Budge, S. M., and Thiemann, G. W.. Distance measures and optimization spaces in quantitative fatty acid signature analysis. Ecology and Evolution, 5:12491262, 2015.CrossRefGoogle ScholarPubMed
[51] Buck, B. and Macaulay, V. A., editors. Maximum Entropy in Action. Oxford University Press, Oxford, 1991.Google Scholar
[52] Buck, D. and Flapan, E.. Predicting knot or catenane type of site-specific recombination products. Journal of Molecular Biology, 374:11861199, 2007.CrossRefGoogle ScholarPubMed
[53] Buck, D. and Flapan, E.. A topological characterization of knots and links arising from site-specific recombination. Journal of Physics A: Mathematical and Theoretical, 40:1237712395, 2007.Google Scholar
[54] Buium, A.. Differential characters of abelian varieties over p-adic fields. Inventiones Mathematicae, 122:309340, 1995.Google Scholar
[55] Buium, A.. Arithmetic analogues of derivations. Journal of Algebra, 198:290– 299, 1997.Google Scholar
[56] Buzas, M. A. and Gibson, T. G.. Species diversity: benthonic Foraminifera in western North Atlantic. Science, 163:7275, 1969.CrossRefGoogle ScholarPubMed
[57] Cannon, J. W. and Floyd, W. J.. What is… Thompson’s group? Notices of the American Mathematical Society, 58(8):11121113, 2011.Google Scholar
[58] Cannon, J. W., Floyd, W. J., and Parry, W. R.. Introductory notes on Richard Thompson’s groups. L’Enseignement Mathématique, 42:215256, 1996.Google Scholar
[59] Carlsson, G.. Topology and data. Bulletin of the American Mathematical Society, 46(2):255308, 2009.Google Scholar
[60] Cathelineau, J.-L.. Sur l’homologie de SL2 à coefficients dans l’action adjointe. Mathematica Scandinavica, 63:5186, 1988.Google Scholar
[61] Cathelineau, J.-L.. Remarques sur les différentielles des polylogarithmes uniformes. Annales de l’Institut Fourier, 46:13271347, 1996.Google Scholar
[62] Čencov, N. N.. Geometry of the “manifold” of a probability distribution (in Russian). Doklady Akademii Nauk SSSR, 158:543546, 1964.Google Scholar
[63] Čencov, N. N.. Statistical Decision Rules and Optimal Inference, volume 53 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1982. Translated from the 1972 Russian edition.Google Scholar
[64] Cerf, R. and Petit, P.. Short proof of Cramér’s theorem in R. American Mathematical Monthly, pages 925–931, December 2011.Google Scholar
[65] Chakravarty, S. R. and Eichhorn, W.. An axiomatic characterization of a generalized index of concentration. Journal of Productivity Analysis, 2:103112, 1991.Google Scholar
[66] Chalmandrier, L., Münkemüller, T., Lavergne, S., and Thuiller, W.. Effects of species’ similarity and dominance on the functional and phylogenetic structure of a plant meta-community. Ecology, 96:143153, 2015.Google Scholar
[67] Chao, A., Chiu, C.-H., and Jost, L.. Phylogenetic diversity measures based on Hill numbers. Philosophical Transactions of the Royal Society B, 365:35993609, 2010.Google Scholar
[68] Cho, S.. Quantales, persistence, and magnitude homology. Preprint arXiv:1910.02905, available at arXiv.org, 2019.Google Scholar
[69] Chuang, J., King, A., and Leinster, T.. On the magnitude of a finite dimensional algebra. Theory and Applications of Categories, 31:6372, 2016.Google Scholar
[70] Chung, K.-S., Chung, W.-S., Nam, S.-T., and Kang, H.-J.. New q-derivative and q-logarithm. International Journal of Theoretical Physics, 33:20192029, 1994.CrossRefGoogle Scholar
[71] Cover, T. M. and Thomas, J. A.. Elements of Information Theory. John Wiley & Sons, New York, 1st edition, 1991.Google Scholar
[72] Cramér, H.. Mathematical Methods of Statistics. Princeton University Press, Princeton, NJ, 1946.Google Scholar
[73] Cressie, N. and Read, T. R. C.. Multinomial goodness-of-fit tests. Journal of the Royal Statistical Society, Series B (Methodological), 46(3):440–464, 1984.CrossRefGoogle Scholar
[74] Csiszár, I.. Generalized cutoff rates and Rényi’s information measures. IEEE Transactions on Information Theory, 41(1):2634, 1995.Google Scholar
[75] Csiszár, I.. Axiomatic characterizations of information measures. Entropy, 10:261273, 2008.CrossRefGoogle Scholar
[76] Csiszár, I. and Shields, P. C.. Information theory and statistics: a tutorial. Foundations and Trends in Communications and Information Theory, 1(4):417528, 2004.Google Scholar
[77] Dahlqvist, F., Danos, V., Garnier, I., and Kammar, O.. Bayesian inversion by ω-complete cone duality. In Desharnais, J. and Jagadeesan, R., editors, 27th International Conference on Concurrency Theory (CONCUR 2016), pages 115. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Saarbrücken, 2016.Google Scholar
[78] Daróczy, Z.. Generalized information functions. Information and Control, 16:36– 51, 1970.Google Scholar
[79] De Boer, P.-T., Kroese, D. P., Mannor, S., and Rubinstein, R. Y.. A tutorial on the cross-entropy method. Annals of Operations Research, 134:1967, 2005.Google Scholar
[80] Dellacherie, C. and Meyer, P.-A.. Probabilities and Potential, C: Potential Theory for Discrete and Continuous Semigroups, volume 151 of North-Holland Mathematics Studies. North-Holland, Amsterdam, 2011.Google Scholar
[81] Deng, L.-Y.. The cross-entropy method: a unified approach to combinatorial optimization, Monte-Carlo simulation, and machine learning. Technometrics, 48:147148, 2006.Google Scholar
[82] Dennis, B. and Patil, G. P.. Profiles of diversity. In Kotz, S. and Johnson, N. L., editors, Encyclopedia of Statistical Sciences, volume 7, pages 292296. John Wiley, New York, 1986.Google Scholar
[83] DeVries, P. J., Murray, D., and Lande, R.. Species diversity in vertical, horizontal and temporal dimensions of a fruit-feeding butterfly community in an Ecuadorian rainforest. Biological Journal of the Linnean Society, 62:343364, 1997.Google Scholar
[84] Downarowicz, T.. Entropy in Dynamical Systems, volume 18 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2011.Google Scholar
[85] Dudley, R. M.. Real Analysis and Probability, volume 74 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2002.Google Scholar
[86] Eguchi, S.. A differential geometric approach to statistical inference on the basis of contrast functionals. Hiroshima Mathematical Journal, 15:341391, 1985.CrossRefGoogle Scholar
[87] Eguchi, S.. Geometry of minimum contrast. Hiroshima Mathematical Journal, 22:641647, 1992.Google Scholar
[88] Elbaz-Vincent, P. and Gangl, H.. On poly(ana)logs I. Compositio Mathematica, 130:161214, 2002.CrossRefGoogle Scholar
[89] Elbaz-Vincent, P. and Gangl, H.. Finite polylogarithms, their multiple analogues and the Shannon entropy. In Nielsen, F. and Barbaresco, F., editors, Geometric Science of Information 2015, volume 9389 of Lecture Notes in Computer Science, pages 277285. Springer, Cham, 2015.Google Scholar
[90] Ellingsen, K. E.. Biodiversity of a continental shelf soft-sediment macrobenthos community. Marine Ecology Progress Series, 218:115, 2001.Google Scholar
[91] Ellison, A. M.. Partitioning diversity. Ecology, 91:19621963, 2010.Google Scholar
[92] Erdős, P.. On the distribution function of additive functions. Annals of Mathematics, 47:120, 1946.Google Scholar
[93] Erdős, P.. On the distribution of additive arithmetical functions and on some related problems. Rendiconti del Seminario Matematico e Fisico di Milano, 27:4549, 1957.Google Scholar
[94] Ernst, T.. A Comprehensive Treatment of q-Calculus. Birkhäuser, Basel, 2012.Google Scholar
[95] Everest, G. and Ward, T.. Heights of Polynomials and Entropy in Algebraic Dynamics. Universitext. Springer, London, 1999.Google Scholar
[96] Faddeev, D. K.. On the concept of entropy of a finite probabilistic scheme (in Russian). Uspekhi Matematicheskikh Nauk, 11:227231, 1956.Google Scholar
[97] Faith, D. P.. Conservation evaluation and phylogenetic diversity. Biological Conservation, 61:110, 1992.Google Scholar
[98] Falconer, K.. Fractal Geometry. John Wiley & Sons, Chichester, 1990.Google Scholar
[99] Feinstein, A.. Foundations of Information Theory. McGraw-Hill, New York, 1958.Google Scholar
[100] Fenchel, W.. On conjugate convex functions. Canadian Journal of Mathematics, 1:7377, 1949.CrossRefGoogle Scholar
[101] Fermi, E.. Thermodynamics. Dover Books on Physics. Dover, New York, 1956.Google Scholar
[102] Fernández-González, C., Palazuelos, C., and Pérez-García, D.. The natural rearrangement invariant structure on tensor products. Journal of Mathematical Analysis and Applications, 343:4047, 2008.Google Scholar
[103] Feynman, R. P.. Negative probability. In Hiley, B. and Peat, D. F., editors, Quantum Implications: Essays in Honour of David Bohm, pages 235248. Routledge, London, 1987.Google Scholar
[104] Fiore, M. and Leinster, T.. An abstract characterization of Thompson’s group F. Semigroup Forum, 80:325340, 2010.Google Scholar
[105] Fiore, T. M., Lück, W., and Sauer, R.. Euler characteristics of categories and homotopy colimits. Documenta Mathematica, 16:301354, 2011.Google Scholar
[106] Fiore, T. M., Lück, W., and Sauer, R.. Finiteness obstructions and Euler characteristics of categories. Advances in Mathematics, 226:23712469, 2011.Google Scholar
[107] Fodor, J. C. and Marichal, J.-L.. On nonstrict means. Aequationes Mathematicae, 54:308327, 1997.Google Scholar
[108] Folland, G. B.. Real Analysis: Modern Techniques and Their Applications. John Wiley & Sons, New York, 2nd edition, 1999.Google Scholar
[109] Forte, B. and Ng, C. T.. On a characterization of the entropies of degree β. Utilitas Mathematica, 4:193205, 1973.Google Scholar
[110] Fréchet, M.. Pri la funkcia ekvacio f (x + y) = f (x) + f (y). L’Enseignement Mathématique, 15:390393, 1913.Google Scholar
[111] Freyd, P.. Algebraic real analysis. Theory and Applications of Categories, 20:215306, 2008.Google Scholar
[112] Fruth, B., Hickey, J. R., André, C., Furuichi, T., Hart, J., Hart, T., Kuehl, H., Maisels, F., Nackoney, J., Reinartz, G., Sop, T., Thompson, J., and Williamson, E. A.. Pan paniscus (errata version published in 2016). IUCN Red List of Threatened Species e.T15932A102331567, 2016.Google Scholar
[113] Furuichi, S.. Uniqueness theorems for Tsallis entropy and Tsallis relative entropy. IEEE Transactions on Information Theory, 51:36383645, 2005.Google Scholar
[114] Gács, P.. Quantum algorithmic entropy. Journal of Physics A: Mathematical and General, 34:68596880, 2001.Google Scholar
[115] Ghrist, R.. Barcodes: the persistent topology of data. Bulletin of the American Mathematical Society, 45:6175, 2008.Google Scholar
[116] Ghrist, R.. Elementary Applied Topology. Createspace, 2014.Google Scholar
[117] Gimperlein, H. and Goffeng, M.. On the magnitude function of domains in Euclidean space. American Journal of Mathematics, to appear, 2021.Google Scholar
[118] Gini, C.. Variabilità e mutabilità. Studi economico-giuridici della facoltà di Giurisprodenza della Regia Università di Cagliari, Anno III, Parte II, 1912.Google Scholar
[119] Gomi, K.. Magnitude homology of geodesic space. Preprint arXiv:1902.07044, available at arXiv.org, 2019.Google Scholar
[120] Gomi, K.. Smoothness filtration of the magnitude complex. Forum Mathematicum, 32(3):625639, 2020.Google Scholar
[121] Good, I. J.. Some terminology and notation in information theory. Proceedings of the IEE, Part C: Monographs, 103(3):200204, 1956.Google Scholar
[122] Good, I. J.. Maximum entropy for hypothesis formulation, especially for multidimensional contingency tables. Annals of Mathematical Statistics, 34(3):911– 934, 1963.Google Scholar
[123] Good, I. J.. Diversity as a concept and its measurement: comment. Journal of the American Statistical Association, 77(379):561563, 1982.Google Scholar
[124] Gould, M.. Coherence for Categorified Operadic Theories. PhD thesis, University of Glasgow, 2008.Google Scholar
[125] Govc, D. and Hepworth, R.. Persistent magnitude. Journal of Pure and Applied Algebra, 225(3):106517, 2021.Google Scholar
[126] Grabisch, M., Marichal, J.-L., Mesiar, R., and Pap, E.. Aggregation Functions, volume 127 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2009.Google Scholar
[127] Gray, A.. Tubes, volume 221 of Progress in Mathematics. Springer, Basel, 2nd edition, 2004.Google Scholar
[128] Greco, R., Di Nardo, A., and Aantonastaso, G.. Resilience and entropy as indices of robustness of water distribution networks. Journal of Hydroinformatics, 14:761771, 2012.Google Scholar
[129] Grendár, M. and Niven, R. K.. The Pólya information divergence. Information Sciences, 180:41894194, 2010.CrossRefGoogle Scholar
[130] Grimmett, G. and Stirzaker, D.. Probability and Random Processes. Oxford University Press, Oxford, 3rd edition, 2001.Google Scholar
[131] Gromov, M.. Metric Structures for Riemannian and Non-Riemannian Spaces. Birkhäuser, Boston, MA, 2001.Google Scholar
[132] Gromov, M.. In a search for a structure, part 1: On entropy. Preprint, 2012.Google Scholar
[133] Gu, Y.. Graph magnitude homology via algebraic Morse theory. Preprint arXiv:1809.07240, available at arXiv.org, 2018.Google Scholar
[134] Hadwiger, H.. Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer, Berlin, 1957.Google Scholar
[135] Hales, T. C.. The NSA back door to NIST. Notices of the American Mathematical Society, 61(2):190192, 2014.Google Scholar
[136] Hannah, L. and Kay, J.. Concentration in the Modern Industry: Theory, Measurement, and the U.K. Experience. MacMillan, London, 1977.Google Scholar
[137] Hardy, G., Littlewood, J. E., and Pólya, G.. Inequalities. Cambridge University Press, Cambridge, 2nd edition, 1952.Google Scholar
[138] Harte, J.. Maximum Entropy and Ecology: A Theory of Abundance, Distribution, and Energetics. Oxford Series in Ecology and Evolution. Oxford University Press, Oxford, 2011.Google Scholar
[139] Haszpra, T. and Tél, T.. Topological entropy: a Lagrangian measure of the state of the free atmosphere. Journal of the Atmospheric Sciences, 70:40304040, 2013.Google Scholar
[140] Hatcher, A.. Algebraic Topology. Cambridge University Press, Cambridge, 2002.Google Scholar
[141] Havrda, J. and Charvát, F.. Quantification method of classification processes: concept of structural α-entropy. Kybernetika, 3:3035, 1967.Google Scholar
[142] Hennessy, M. and Milner, R.. Algebraic laws for nondeterminism and concurrency. Journal of the Association for Computing Machinery, 32(1):137161, 1985.Google Scholar
[143] Hepworth, R.. Magnitude cohomology. Preprint arXiv:1807.06832, available at arXiv.org, 2018.Google Scholar
[144] Hepworth, R. and Willerton, S.. Categorifying the magnitude of a graph. Homology, Homotopy and Applications, 19(2):3160, 2017.Google Scholar
[145] Hey, J.. The mind of the species problem. Trends in Ecology and Evolution, 16(7):326329, 2001.Google Scholar
[146] Hill, M. O.. Diversity and evenness: a unifying notation and its consequences. Ecology, 54(2):427432, 1973.Google Scholar
[147] Hobson, A.. A new theorem of information theory. Journal of Statistical Physics, 1(3):383391, 1969.Google Scholar
[148] Huffman, D. A.. A method for the construction of minimum-redundancy codes. Proceedings of the IRE, 40:10981101, 1952.Google Scholar
[149] Humle, T., Maisels, F., Oates, J. F., Plumptre, A., and Williamson, E. A.. Pan troglodytes (errata version published in 2016). IUCN Red List of Threatened Species e.T15933A102326672, 2016.Google Scholar
[150] Hurlbert, S. H.. The nonconcept of species diversity: a critique and alternative parameters. Ecology, 52(4):577586, 1971.Google Scholar
[151] Ives, A. R.. Diversity and stability in ecological communities. In May, R. M. and McLean, A. R., editors, Theoretical Ecology: Principles and Applications. Oxford University Press, Oxford, 2007.Google Scholar
[152] Izsák, J. and Szeidl, L.. Quadratic diversity: its maximization can reduce the richness of species. Environmental and Ecological Statistics, 9:423430, 2002.Google Scholar
[153] Jaccard, P.. Nouvelles recherches sur la distribution florale. Bulletin de la Société Vaudoise des Sciences Naturelles, 40:223270, 1908.Google Scholar
[154] Jackson, A.. Comme appelé du néant – as if summoned from the void: the life of Alexandre Grothendieck. Notices of the American Mathematical Society, 51(9):10381056, 2004.Google Scholar
[155] Jackson, F. H.. On q-functions and a certain difference operator. Transactions of the Royal Society of Edinburgh, 46:253281, 1908.Google Scholar
[156] Jaynes, E. T.. Where do we stand on maximum entropy? In Levine, R. D. and Tribus, M., editors, The Maximum Entropy Formalism, pages 15118. MIT Press, Cambridge, MA, 1979.Google Scholar
[157] Jaynes, E. T.. Probability Theory: The Logic of Science. Cambridge University Press, Cambridge, 2003.Google Scholar
[158] Jeffreys, H.. Theory of Probability. Clarendon Press, Oxford, 1st edition, 1939.Google Scholar
[159] Jeffreys, H.. An invariant form for the prior probability in estimation problems. Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences, 186:453461, 1946.Google Scholar
[160] Jeziorski, A., Tanentzap, A. J., Yan, N. D., Paterson, A. M., Palmer, M. E., Korosi, J. B., Rusak, J. A., Arts, M. T., Keller, W., Ingram, R., Cairns, A., and Smol, J. P.. The jellification of north temperate lakes. Proceedings of the Royal Society B, 282:20142449, 2015.Google Scholar
[161] Johnson, J. L.. Use of nucleic-acid homologies in taxonomy of anaerobic bacteria. International Journal of Systematic Bacteriology, 23(4):308315, 1973.CrossRefGoogle Scholar
[162] Johnson, O.. Information Theory and the Central Limit Theorem. Imperial College Press, London, 2004.Google Scholar
[163] Johnstone, P. T.. Topos Theory. Academic Press, London, 1977.Google Scholar
[164] Jones, G. A. and Jones, J. M.. Information and Coding Theory. Springer Undergraduate Mathematics Series. Springer, London, 2000.Google Scholar
[165] Jost, J.. Riemannian Geometry and Geometric Analysis. Universitext. Springer, Berlin, 7th edition, 2017.Google Scholar
[166] Jost, L.. Entropy and diversity. Oikos, 113:363375, 2006.Google Scholar
[167] Jost, L.. Partitioning diversity into independent alpha and beta components. Ecology, 88(10):24272439, 2007.Google Scholar
[168] Jost, L.. GST and its relatives do not measure differentiation. Molecular Ecology, 17:40154026, 2008.Google Scholar
[169] Jost, L.. Mismeasuring biological diversity: response to Hoffmann and Hoffmann (2008). Ecological Economics, 68:925928, 2009.Google Scholar
[170] Jost, L.. Independence of alpha and beta diversities. Ecology, 91:19691974, 2010.Google Scholar
[171] Jost, L., DeVries, P., Walla, T., Greeney, H., Chao, A., and Ricotta, C.. Partitioning diversity for conservation analyses. Diversity and Distributions, 16:6576, 2010.Google Scholar
[172] Joyal, A., Nielsen, M., and Winskel, G.. Bisimulation from open maps. Information and Computation, 127:164185, 1996.Google Scholar
[173] Joyce, J. M.. Kullback–Leibler divergence. In Lovric, M., editor, International Encyclopedia of Statistical Science, pages 720722. Springer, Berlin, 2011.CrossRefGoogle Scholar
[174] Jubin, B.. On the magnitude homology of metric spaces. Preprint arXiv:1803.05062, available at arXiv.org, 2018.Google Scholar
[175] Kac, V. and Cheung, P.. Quantum Calculus. Universitext. Springer, New York, 2002.Google Scholar
[176] Kaneta, R. and Yoshinaga, M.. Magnitude homology of metric spaces and order complexes. Preprint arXiv:1803.04247, available at arXiv.org, 2018.Google Scholar
[177] Kannappan, P. and Ng, C. T.. Measurable solutions of functional equations related to information theory. Proceedings of the American Mathematical Society, 38:303310, 1973.Google Scholar
[178] Kannappan, P. and Ng, C. T.. On functional equations connected with directed divergence, inaccuracy and generalized directed divergence. Pacific Journal of Mathematics, 54(1):157167, 1974.Google Scholar
[179] Kannappan, P. and Rathie, P. N.. On a characterization of directed divergence. Information and Control, 22:163171, 1973.Google Scholar
[180] Kanter, M.. Discrimination distance bounds and statistical applications. Probability Theory and Related Fields, 86:403422, 1990.Google Scholar
[181] Karp, R. M.. Reducibility among combinatorial problems. In Miller, R. E. and Thatcher, J. W., editors, Complexity of Computer Computations, pages 85103. Plenum Press, New York, 1972.Google Scholar
[182] Kass, R. E. and Wasserman, L.. The selection of prior distributions by formal rules. Journal of the American Statistical Association, 91:13431370, 1996.Google Scholar
[183] Kátai, I.. A remark on additive arithmetical functions. Annales Universitatis Scientiarum Budapestinensis de Rolando Eőtvős Nominatae, Sectio Mathematica, 10:8183, 1967.Google Scholar
[184] Kelly, G. M.. Basic Concepts of Enriched Category Theory, volume 64 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1982. Also Reprints in Theory and Applications of Categories 10:1–136, 2005.Google Scholar
[185] Kelly, G. M.. On the operads of J. P. May. Reprints in Theory and Applications of Categories, 13:113, 2005.Google Scholar
[186] Kerridge, D. F.. Inaccuracy and inference. Journal of the Royal Statistical Society, Series B (Methodological), 23:184–194, 1961.Google Scholar
[187] Keylock, C. J.. Simpson diversity and the Shannon–Wiener index as special cases of a generalized entropy. Oikos, 109(1):203207, 2005.Google Scholar
[188] Khinchin, A. I.. Mathematical Foundations of Information Theory. Dover, New York, 1957.Google Scholar
[189] Khovanov, M.. A categorification of the Jones polynomial. Duke Mathematical Journal, 101:359426, 2000.Google Scholar
[190] Klain, D. A. and Rota, G.-C.. Introduction to Geometric Probability. Lezioni Lincee. Cambridge University Press, Cambridge, 1997.Google Scholar
[191] Kock, J.. Frobenius Algebras and 2D Topological Quantum Field Theories, volume 59 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 2003.Google Scholar
[192] Kolmogorov, A. N.. Sur la notion de la moyenne. Atti della Accademia Nazionale dei Lincei, Rendiconti, VI Serie, 12:388391, 1930.Google Scholar
[193] Kolmogorov, A. N.. On certain asymptotic characteristics of completely bounded metric spaces. Doklady Akademii Nauk SSSR, 108:385388, 1956.Google Scholar
[194] Kolmogorov, A. N.. On the notion of mean. In Tikhomirov, V. M., editor, Selected Works of A. N. Kolmogorov, volume I: Mathematics and Mechanics, volume 25 of Mathematics and Its Applications (Soviet Series), pages 144–146. Springer Science and Business Media, Dordrecht, 1991.Google Scholar
[195] Kontsevich, M.. The 11/2-logarithm. Private note, 1995. Reprinted as appendix of [88].Google Scholar
[196] Kontsevich, M.. Operads and motives in deformation quantization. Letters in Mathematical Physics, 48(1):3572, 1999.Google Scholar
[197] Kovačević, M., Stanojević, I., and Šenk, V.. On the entropy of couplings. Information and Computation, 242:369382, 2015.Google Scholar
[198] Kullback, S.. Information Theory and Statistics. John Wiley & Sons, New York, 1959.Google Scholar
[199] Kullback, S.. The Kullback–Leibler distance. American Statistician, 41(4):340341, 1987.Google Scholar
[200] Laakso, M. and Taagepera, R.. “Effective” number of parties: a measure with application to West Europe. Comparative Political Studies, 12:327, 1979.Google Scholar
[201] Lambek, J.. Deductive systems and categories II: standard constructions and closed categories. In Hilton, P., editor, Category Theory, Homology Theory and their Applications, I (Battelle Institute Conference, Seattle, 1968), volume 86 of Lecture Notes in Mathematics. Springer, Berlin, 1969.Google Scholar
[202] Lauritzen, S. L.. Statistical manifolds. In Amari, S.-I., Barndorff-Nielsen, O. E., Kass, R. E., Lauritzen, S. L., and Rao, C. R., editors, Differential Geometry in Statistical Inference, volume 10 of Lecture Notes – Monograph Series, pages 163216. Institute of Mathematical Statistics, Hayward, CA, 1987.Google Scholar
[203] Lawvere, F. W.. Functorial Semantics of Algebraic Theories. PhD thesis, Columbia University, 1963. Also Reprints in Theory and Applications of Categories 5:1–121, 2004.Google Scholar
[204] Lawvere, F. W.. Metric spaces, generalized logic and closed categories. Rendiconti del Seminario Matematico e Fisico di Milano, XLIII:135–166, 1973. Also Reprints in Theory and Applications of Categories 1:1–37, 2002.Google Scholar
[205] Lawvere, F. W.. State categories, closed categories, and the existence semi-continuous entropy functions. IMA Preprint Series 86, Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, MN, 1984.Google Scholar
[206] Lee, P. M.. On the axioms of information theory. Annals of Mathematical Statistics, 35:415418, 1964.Google Scholar
[207] Leinster, T.. Higher Operads, Higher Categories, volume 298 of London Mathematical Society Lecture Notes Series. Cambridge University Press, Cambridge, 2004.Google Scholar
[208] Leinster, T.. Are operads algebraic theories? Bulletin of the London Mathematical Society, 38:233238, 2006.Google Scholar
[209] Leinster, T.. General self-similarity: an overview. In Paunescu, L., Harris, A., Fukui, T., and Koike, S., editors, Real and Complex Singularities. World Scientific, Singapore, 2007.Google Scholar
[210] Leinster, T.. The Euler characteristic of a category. Documenta Mathematica, 13:2149, 2008.Google Scholar
[211] Leinster, T.. A maximum entropy theorem with applications to the measurement of biodiversity. Preprint arXiv:0910.0906, available at arXiv.org, 2009.Google Scholar
[212] Leinster, T.. A general theory of self-similarity. Advances in Mathematics, 226:29353017, 2011.Google Scholar
[213] Leinster, T.. Integral geometry for the 1-norm. Advances in Applied Mathematics, 49:8196, 2012.Google Scholar
[214] Leinster, T.. A multiplicative characterization of the power means. Bulletin of the London Mathematical Society, 44:106112, 2012.Google Scholar
[215] Leinster, T.. Notions of Möbius inversion. Bulletin of the Belgian Mathematical Society, 19:911935, 2012.Google Scholar
[216] Leinster, T.. The magnitude of metric spaces. Documenta Mathematica, 18:857– 905, 2013.Google Scholar
[217] Leinster, T.. The magnitude of a graph. Mathematical Proceedings of the Cambridge Philosophical Society, 166:247264, 2019.Google Scholar
[218] Leinster, T.. A short characterization of relative entropy. Journal of Mathematical Physics, 60(2):023302, 2019.Google Scholar
[219] Leinster, T.. Entropy modulo a prime. Communications in Number Theory and Physics, to appear, 2021.Google Scholar
[220] Leinster, T. and Cobbold, C. A.. Measuring diversity: the importance of species similarity. Ecology, 93:477489, 2012.Google Scholar
[221] Leinster, T. and Meckes, M.. Maximizing diversity in biology and beyond. Entropy, 18(88), 2016.Google Scholar
[222] Leinster, T. and Meckes, M.. The magnitude of a metric space: from category theory to geometric measure theory. In Gigli, N., editor, Measure Theory in Non-Smooth Spaces, pages 156193. De Gruyter Open, Warsaw, 2017.Google Scholar
[223] Leinster, T. and Roff, E.. The maximum entropy of a metric space. Quarterly Journal of Mathematics, to appear, 2021.Google Scholar
[224] Leinster, T. and Shulman, M.. Magnitude homology of enriched categories and metric spaces. Algebraic & Geometric Topology, to appear, 2021.Google Scholar
[225] Leinster, T. and Willerton, S.. On the asymptotic magnitude of subsets of Euclidean space. Geometriae Dedicata, 164:287310, 2013.Google Scholar
[226] Lemey, P., Salemi, M., and Vandamme, A.-M.. The Phylogenetic Handbook. Cambridge University Press, Cambridge, 2nd edition, 2009.Google Scholar
[227] Lesnick, M.. Studying the shape of data using topology. The Institute Letter, pages 10–11, Summer 2013. Institute for Advanced Study, Princeton, NJ.Google Scholar
[228] Lin, Z.. Are groups algebras over an operad? Mathematics Stack Exchange, 2013. Available at https://math.stackexchange.com/q/366371.Google Scholar
[229] Lindhard, J. and Nielsen, V.. Studies in statistical dynamics. Matematisk-Fysiske Meddelelser: Kongelige Danske Videnskabernes Selskab, 38:142, 1971.Google Scholar
[230] Lindley, D. V.. Making Decisions. Wiley, London, 2nd edition, 1985.Google Scholar
[231] Lindley, D. V.. Understanding Uncertainty. Wiley, Hoboken, NJ, 2006.Google Scholar
[232] Linnik, Y. V.. An information-theoretic proof of the central limit theorem with the Lindeberg condition. Theory of Probability and its Applications, 4(3):288299, 1959.Google Scholar
[233] Littlewood, J. E.. Lectures on the Theory of Functions. Oxford University Press, London, 1944.Google Scholar
[234] Loday, J.-L. and Vallette, B.. Algebraic Operads, volume 346 of Grundlehren der Mathematischen Wissenschaften. Springer, Berlin, 2012.Google Scholar
[235] Lusin, N.. Sur les propriétés des fonctions mesurables. Comptes Rendus Hebdo-madaires des Séances de l’Académie des Sciences, 154:16881690, 1912.Google Scholar
[236] Mac Lane, S.. Categories for the Working Mathematician. Graduate Texts in Mathematics 5. Springer, New York, 1971.Google Scholar
[237] MacArthur, R. H.. Patterns of species diversity. Biological Reviews, 40:510533, 1965.Google Scholar
[238] MacKay, D. J. C.. Information Theory, Inference and Learning Algorithms. Cambridge University Press, Cambridge, 2003.Google Scholar
[239] Mackey, M. C. and Maini, P. K.. What has mathematics done for biology? Bulletin of Mathematical Biology, 77:735738, 2015.Google Scholar
[240] Magurran, A. E.. Measuring Biological Diversity. Blackwell, Oxford, 2004.Google Scholar
[241] Maisels, F., Bergl, R. A., and Williamson, E. A.. Gorilla gorilla (errata version published in 2016). IUCN Red List of Threatened Species e.T9404A102330408, 2016.Google Scholar
[242] Manes, E.. Algebraic Theories, volume 26 of Graduate Texts in Mathematics. Springer, Berlin, 1976.Google Scholar
[243] Manke, T., Demetrius, L., and Vingron, M.. An entropic characterization of protein interaction networks and cellular robustness. Journal of the Royal Society Interface, 3:843850, 2006.Google Scholar
[244] Markl, M., Shnider, S., and Stasheff, J.. Operads in Algebra, Topology and Physics, volume 96 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2002.Google Scholar
[245] Máté, A.. A new proof of a theorem of P. Erdős. Proceedings of the American Mathematical Society, 18:159162, 1967.Google Scholar
[246] Mather, A. E., Matthews, L., Mellor, D. J., Reeve, R., Denwood, M. J., Boerlin, P., Reid-Smith, R. J., Brown, D. J., Coia, J. E., Browning, L. M., Haydon, D. T., and Reid, S. W. J.. An ecological approach to assessing the epidemiology of antimicrobial resistance in animal and human populations. Proceedings of the Royal Society B, 279:16301639, 2012.Google Scholar
[247] May, J. P.. The Geometry of Iterated Loop Spaces, volume 271 of Lecture Notes in Mathematics. Springer, Berlin, 1972.Google Scholar
[248] May, J. P.. Definitions: operads, algebras and modules. In Loday, J.-L., Stasheff, J., and Voronov, A., editors, Operads: Proceedings of Renaissance Conferences, volume 202 of Contemporary Mathematics. American Mathematical Society, Providence, RI, 1997.Google Scholar
[249] Mayden, R. L.. A hierarchy of species concepts: the denouement in the saga of the species problem. In Claridge, M. F., Dawah, H. A., and Wilson, M. R., editors, Species: The Units of Biodiversity, pages 381424. Chapman & Hall, London, 1997.Google Scholar
[250] Meckes, M. W.. Positive definite metric spaces. Positivity, 17:733757, 2013.Google Scholar
[251] Meckes, M. W.. Magnitude, diversity, capacities, and dimensions of metric spaces. Potential Analysis, 42:549572, 2015.Google Scholar
[252] Meckes, M. W.. On the magnitude and intrinsic volumes of a convex body in Euclidean space. Mathematika, 66:343355, 2020.Google Scholar
[253] Mendes, R. S., Evangelista, L. R., Thomaz, S. M., Agostinho, A. A., and Gomes, L. C.. A unified index to measure ecological diversity and species rarity. Ecography, 31:450456, 2008.Google Scholar
[254] Mittermeier, R. A., Ganzhorn, J. U., Konstant, W. R., Glander, K., Tattersall, I., Groves, C. P., Rylands, A. B., Hapke, A., Ratsimbazafy, J., Mayor, M. I., Louis, E. E. Jr., Rumpler, Y., Schwitzer, C., and Rasoloarison, R. M.. Lemur diversity in Madagascar. International Journal of Primatology, 29:16071656, 2008.Google Scholar
[255] Morvan, J.-M.. Generalized Curvatures. Springer, Berlin, 2008.Google Scholar
[256] Motzkin, T. S. and Straus, E. G.. Maxima for graphs and a new proof of a theorem of Turán. Canadian Journal of Mathematics, 17:533540, 1965.Google Scholar
[257] Moulin, H. J.. Fair Division and Collective Welfare. MIT Press, Cambridge, MA, 2003.Google Scholar
[258] Mureşan, M.. A Concrete Approach to Classical Analysis. CMS Books in Mathematics. Springer, New York, 2009.Google Scholar
[259] Nagendra, H.. Opposite trends in response for the Shannon and Simpson indices of landscape diversity. Applied Geography, 22:175186, 2002.Google Scholar
[260] Nagumo, M.. Über eine Klasse der Mittelwerte. Japanese Journal of Mathematics, 7:7179, 1930.Google Scholar
[261] Nater, A., Mattle-Greminger, M. P., Nurcahyo, A., Nowak, M. G., de Manuel, M., Desai, T., Groves, C., Pybus, M., Bilgin Sonay, T., Roos, C., Lameira, A. R., Wich, S. A., Askew, J., Davila-Ross, M., Fredriksson, G., de Valles, G., Casals, F., Prado-Martinez, J., Goossens, B., Verschoor, E. J., Warren, K. S., Singleton, I., Marques, D. A., Pamungkas, J., Perwitasari-Farajallah, D., Rianti, P., Tuuga, A., Gut, I. G., Gut, M., Orozco-terWengel, P., van Schaik, C. P., Bertranpetit, J., Anisimova, M., Scally, A., Marques-Bonet, T., Meijaard, E., and Krützen, M.. Morphometric, behavioral, and genomic evidence for a new orangutan species. Current Biology, 27(22):3487–3498, 2017.Google Scholar
[262] Nee, S.. More than meets the eye. Nature, 429:804805, 2004.Google Scholar
[263] Nicolau, M., Levine, A. J., and Carlsson, G.. Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival. Proceedings of the National Academy of Sciences of the USA, 108(17):72657270, 2011.Google Scholar
[264] Noguchi, K.. The Euler characteristic of acyclic categories. Kyushu Journal of Mathematics, 65:8599, 2011.Google Scholar
[265] Noguchi, K.. Euler characteristics of categories and barycentric subdivision. Münster Journal of Mathematics, 6:85116, 2013.Google Scholar
[266] Noguchi, K.. The zeta function of a finite category. Documenta Mathematica, 18:12431274, 2013.Google Scholar
[267] Nygaard, M. and Winskel, G.. Domain theory for concurrency. Theoretical Computer Science, 316:153190, 2004.Google Scholar
[268] OECD. Handbook of Biodiversity Valuation: A Guide for Policy Makers. Organisation for Economic Co-operation and Development, Paris, 2002.Google Scholar
[269] Otter, N.. Magnitude meets persistence. Homology theories for filtered simplicial sets. Preprint arXiv:1807.01540, available at arXiv.org, 2018.Google Scholar
[270] Parry, W.. Entropy and Generators in Ergodic Theory. W. A. Benjamin, New York, 1969.Google Scholar
[271] Patil, G. P.. Diversity profiles. In El-Shaarawi, A. H. and Piegorsch, W. W., editors, Encyclopedia of Environmetrics. John Wiley & Sons, Chichester, 2002.Google Scholar
[272] Patil, G. P. and Taillie, C.. A study of diversity profiles and orderings for a bird community in the vicinity of Colstrip, Montana. In Patil, G. P. and Rosenzweig, M., editors, Contemporary Quantitative Ecology and Related Ecometrics, pages 2348. International Co-operative Publishing House, Fairland, MD, 1979.Google Scholar
[273] Patil, G. P. and Taillie, C.. Diversity as a concept and its measurement. Journal of the American Statistical Association, 77(379):548561, 1982.Google Scholar
[274] Pavlovic, D.. Quantitative concept analysis. In Domenach, F., Ignatov, D. I., and Poelmans, J., editors, Formal Concept Analysis. ICFCA 2012, volume 7278 of Lecture Notes in Computer Science, pages 260277. Springer, Berlin, 2012.Google Scholar
[275] Pavoine, S. and Bonsall, M. B.. Biological diversity: distinct distributions can lead to the maximization of Rao’s quadratic entropy. Theoretical Population Biology, 75:153163, 2009.Google Scholar
[276] Pavoine, S., Ollier, S., and Pontier, D.. Measuring diversity from dissimilarities with Rao’s quadratic entropy: Are any dissimilarities suitable? Theoretical Population Biology, 67:231239, 2005.Google Scholar
[277] Peet, R. K.. The measurement of species diversity. Annual Review of Ecology and Systematics, 5:285307, 1974.Google Scholar
[278] Pennec, X.. Barycentric subspace analysis on manifolds. Annals of Statistics, 46:27112746, 2018.Google Scholar
[279] Peres, A., Scudo, P. F., and Terno, D. R.. Quantum entropy and special relativity. Physical Review Letters, 88(23):230402, 2002.Google Scholar
[280] Petchey, O. L. and Gaston, K. J.. Functional diversity: back to basics and looking forward. Ecology Letters, 9:741758, 2006.Google Scholar
[281] Petz, D.. Characterization of the relative entropy of states of matrix algebras. Acta Mathematica Hungarica, 59:449455, 1992.Google Scholar
[282] Pielou, E. C.. Ecological Diversity. John Wiley & Sons, New York, 1975.Google Scholar
[283] Pielou, E. C.. Mathematical Ecology. John Wiley & Sons, New York, 2nd edition, 1977.Google Scholar
[284] Plumptre, A., Robbins, M., and Williamson, E. A.. Gorilla beringei (errata version published in 2016). IUCN Red List of Threatened Species e.T39994A102325702, 2016.Google Scholar
[285] Rao, C. R.. Information and the accuracy attainable in the estimation of statistical parameters. Bulletin of the Calcutta Mathematical Society, 37:8189, 1945.Google Scholar
[286] Rao, C. R.. Diversity and dissimilarity coefficients: a unified approach. Theoretical Population Biology, 21:2443, 1982.Google Scholar
[287] Rao, C. R.. Diversity: its measurement, decomposition, apportionment and analysis. Sankhyā: The Indian Journal of Statistics, 44(1):122, 1982.Google Scholar
[288] Rao, C. R.. Differential metrics in probability spaces. In Differential Geometry in Statistical Inference, volume 10 of Lecture Notes – Monograph Series, pages 217–240. Institute of Mathematical Statistics, Hayward, CA, 1987.Google Scholar
[289] Rathie, P. N. and Kannappan, P.. A directed-divergence function of type β. Information and Control, 20:3845, 1972.Google Scholar
[290] Ratnaparkhi, A.. Learning to parse natural language with maximum entropy models. Machine Learning, 34:151175, 1999.Google Scholar
[291] Reed, M. C.. Mathematical biology is good for mathematics. Notices of the American Mathematical Society, 62(10):11721176, 2015.Google Scholar
[292] Reem, D.. Remarks on the Cauchy functional equation and variations of it. Aequationes Mathematicae, 91:237264, 2017.Google Scholar
[293] Reeve, R., Leinster, T., Cobbold, C. A., Thompson, J., Brummitt, N., Mitchell, S. N., and Matthews, L.. How to partition diversity. Preprint arXiv:1404.6520v3, available at arXiv.org, 2016.Google Scholar
[294] Rényi, A.. On measures of entropy and information. In Neyman, J., editor, Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, volume 1, pages 547561. University of California Press, Berkeley, CA, 1961.Google Scholar
[295] Rényi, A.. Probability Theory. North-Holland, Amsterdam, 1970.Google Scholar
[296] Ricotta, C. and Szeidl, L.. Towards a unifying approach to diversity measures: bridging the gap between the Shannon entropy and Rao’s quadratic index. Theoretical Population Biology, 70:237243, 2006.Google Scholar
[297] Riehl, E.. Category Theory in Context. Dover, New York, 2016.Google Scholar
[298] Robert, C. P., Chopin, N., and Rousseau, J.. Harold Jeffreys’s Theory of Probability revisited. Statistical Science, 24:141172, 2009.Google Scholar
[299] Rockafellar, R. T.. Convex Analysis. Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1970.Google Scholar
[300] Roman, S.. Field Theory, volume 158 of Graduate Texts in Mathematics. Springer, New York, 2nd edition, 2006.Google Scholar
[301] Rota, G.-C.. On the foundations of combinatorial theory I: theory of Möbius functions. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 2:340368, 1964.Google Scholar
[302] Routledge, R. D.. Diversity indices: which ones are admissible? Journal of Theoretical Biology, 76:503515, 1979.Google Scholar
[303] Sason, I.. Entropy bounds for discrete random variables via maximal coupling. IEEE Transactions on Information Theory, 59(11):71187131, 2013.Google Scholar
[304] Schanuel, S. H.. Negative sets have Euler characteristic and dimension. In Carboni, A., Pedicchio, M. C., and Rosolini, G., editors, Category Theory (Como, 1990), Lecture Notes in Mathematics 1488, pages 379385. Springer, Berlin, 1991.Google Scholar
[305] Schervish, M. J.. Theory of Statistics. Springer Series in Statistics. Springer, New York, 1995.Google Scholar
[306] Schneider, R.. Convex Bodies: the Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications 44. Cambridge University Press, Cambridge, 2nd edition, 2014.Google Scholar
[307] Schoenberg, I. J.. Metric spaces and positive definite functions. Transactions of the American Mathematical Society, 44:522536, 1938.Google Scholar
[308] Segal, G.. Classifying spaces and spectral sequences. Institut des Hautes Études Scientifiques Publications Mathématiques, 34:105112, 1968.Google Scholar
[309] Shannon, C. E.. A mathematical theory of communication. Bell System Technical Journal, 27:379423, 1948.Google Scholar
[310] Shannon, C. E.. Prediction and entropy of printed English. Bell System Technical Journal, 30:5064, 1951.Google Scholar
[311] Shiino, M.. H-theorem with generalized relative entropies and the Tsallis statistics. Journal of the Physical Society of Japan, 67:36583660, 1998.Google Scholar
[312] Shimatani, K.. The appearance of a different DNA sequence may decrease nucleotide diversity. Journal of Molecular Evolution, 49:810813, 1999.Google Scholar
[313] Shore, J. E. and Johnson, R. W.. Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy. IEEE Transactions on Information Theory, 26(1):2637, 1980.Google Scholar
[314] Simpson, E. H.. Measurement of diversity. Nature, 163:688, 1949.Google Scholar
[315] Singleton, I., Wich, S. A., Nowak, M., and Usher, G.. Pongo abelii (errata version published in 2016). IUCN Red List of Threatened Species e.T39780A102329901, 2016.Google Scholar
[316] Skachkov, V. V., Chepkyi, V. V., Bratchenko, H. D., and Efymchykov, A. N.. Entropy approach to the investigation of information capabilities of adaptive radio engineering system in conditions of intrasystem uncertainty. Radioelectronics and Communications Systems, 58:241249, 2015.Google Scholar
[317] Smith, W. and Grassle, J. F.. Sampling properties of a family of diversity measures. Biometrics, 33(2):283292, 1977.Google Scholar
[318] Solovay, R. M.. A model of set theory in which every set of reals is Lebesgue measurable. Annals of Mathematics, 92:156, 1970.Google Scholar
[319] Solow, A. R. and Polasky, S.. Measuring biological diversity. Environmental and Ecological Statistics, 1:95107, 1994.Google Scholar
[320] Stanley, R. P.. Enumerative Combinatorics, volume 1, volume 49 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1997.Google Scholar
[321] Summers, V.. Torsion in the Khovanov Homology of Links and the Magnitude Homology of Graphs. PhD thesis, North Carolina State University, 2019.Google Scholar
[322] Suyari, H.. On the most concise set of axioms and the uniqueness theorem for Tsallis entropy. Journal of Physics A: Mathematical and General, 35:10731– 10738, 2002.Google Scholar
[323] Tanaka, K.. The Euler characteristic of a bicategory and the product formula for fibered bicategories. Preprint arXiv:1410.0248, available at arXiv.org, 2014.Google Scholar
[324] Taneja, I. J. and Kumar, P.. Relative information of type s, Csiszár’s f -divergence, and information inequalities. Information Sciences, 166:105125, 2004.Google Scholar
[325] Tao, T.. Structure and Randomness: Pages from Year One of a Mathematical Blog. American Mathematical Society, Providence, RI, 2008.Google Scholar
[326] Timme, N., Alford, W., Flecker, B., and Beggs, J. M.. Synergy, redundancy, and multivariate information measures: an experimentalist’s perspective. Journal of Computational Neuroscience, 36:119140, 2014.Google Scholar
[327] Tóthmérész, B.. Comparison of different methods for diversity ordering. Journal of Vegetation Science, 6:283290, 1995.Google Scholar
[328] Tribus, M. and McIrvine, E. C.. Energy and information. Scientific American, 225(3):179190, 1971.Google Scholar
[329] Tronin, S. N.. Abstract clones and operads. Siberian Mathematical Journal, 43(4):746755, 2002.Google Scholar
[330] Tronin, S. N.. Operads and varieties of algebras defined by polylinear identities. Siberian Mathematical Journal, 47(3):555573, 2006.Google Scholar
[331] Tsallis, C.. Possible generalization of Boltzmann–Gibbs statistics. Journal of Statistical Physics, 52:479487, 1988.Google Scholar
[332] Tsallis, C.. What are the numbers that experiments provide? Química Nova, 17(6):468471, 1994.Google Scholar
[333] Tsallis, C.. Generalized entropy-based criterion for consistent testing. Physical Review E, 58:14421445, 1998.Google Scholar
[334] Tuomisto, H.. A diversity of beta diversities: straightening up a concept gone awry. Part 1. Defining beta diversity as a function of alpha and gamma diversity. Ecography, 33:2–22, 2010.Google Scholar
[335] Turnbaugh, P. J., Hamady, M., Yatsunenko, T., Cantarel, B. L., Duncan, A., Ley, R. E., Sogin, M. L., Jones, W. J., Roe, B. A., Affourtit, J. P., Egholm, M., Henrissat, B., Heath, A. C., Knight, R., and Gordon, J. I.. A core gut microbiome in obese and lean twins. Nature, 457:480484, 2009.Google Scholar
[336] Turnbaugh, P. J., Quince, C., Faith, J. J., McHardy, A. C., Yatsunenko, T., Niazi, F., Affourtit, J., Egholm, M., Henrissat, B., Knight, R., and Gordon, J. I.. Organismal, genetic, and transcriptional variation in the deeply sequenced gut microbiomes of identical twins. Proceedings of the National Academy of Sciences of the USA, 107:75037508, 2010.Google Scholar
[337] Tverberg, H.. A new derivation of the information function. Mathematica Scandinavica, 6:297298, 1958.Google Scholar
[338] Twarock, R., Valiunas, M., and Zappa, E.. Orbits of crystallographic embedding of non-crystallographic groups and applications to virology. Acta Crystallographica, A71:569582, 2015.Google Scholar
[339] United Nations, Department of Economic and Social Affairs, Population Division. World population prospects: the 2017 revision. Custom data acquired via website, 2017.Google Scholar
[340] Vajda, I.. Axioms for a-entropy of a generalized probability scheme (in Czech). Kybernetika, 4(2):105112, 1968.Google Scholar
[341] van den Dries, L.. Tame Topology and O-minimal Structures, volume 248 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1998.Google Scholar
[342] Vane-Wright, R. I., Humphries, C. J., and Williams, P. H.. What to protect? – Systematics and the agony of choice. Biological Conservation, 55:235254, 1991.Google Scholar
[343] Varga, R. S. and Nabben, R.. On symmetric ultrametric matrices. In Reichel, L., Ruttan, A., and Varga, R. S., editors, Numerical Linear Algebra, pages 193199. Walter de Gruyter, Berlin, 1993.Google Scholar
[344] Veresoglou, S. D., Powell, J. R., Davison, J., Lekberg, Y., and Rillig, M. C.. The Leinster and Cobbold indices improve inferences about microbial diversity. Fungal Ecology, 11:17, 2014.Google Scholar
[345] Vigneaux, J. P.. Topology of Statistical Systems: A Cohomological Approach to Information Theory. PhD thesis, Université Paris Diderot, 2019.Google Scholar
[346] Wallace, A. R.. Tropical Nature, and Other Essays. MacMillan and Co., London, 1878.Google Scholar
[347] Wang, L., Zhang, M., Jajodia, S., Singhal, A., and Albanese, M.. Modeling network diversity for evaluating the robustness of networks against zero-day attacks. In Kutyłowski, M. and Vaidya, J., editors, , Proceedings of the 19th European Symposium on Research in Computer Security (ESORICS 2014), pages 494511. Springer, Cham, 2014.Google Scholar
[348] Warwick, R. M. and Clarke, K. R.. New ‘biodiversity’ measures reveal a decrease in taxonomic distinctness with increasing stress. Marine Ecology Progress Series, 129:301305, 1995.Google Scholar
[349] Watve, M. G. and Gangal, R. M.. Problems in measuring bacterial diversity and a possible solution. Applied and Environmental Microbiology, 62(11):42994301, 1996.Google Scholar
[350] Wehrl, A.. General properties of entropy. Reviews of Modern Physics, 50(2):221– 260, 1978.Google Scholar
[351] Whittaker, R. H.. Vegetation of the Siskiyou mountains, Oregon and California. Ecological Monographs, 30:279338, 1960.Google Scholar
[352] Whittaker, R. H.. Evolution and measurement of species diversity. Taxon, 21:213251, 1972.Google Scholar
[353] Willerton, S.. Heuristic and computer calculations for the magnitude of metric spaces. Preprint arXiv:0910.5500, available at arXiv.org, 2009.Google Scholar
[354] Willerton, S.. Is this graph of reciprocal power means always convex? Math-Overflow, 2014. Available at https://mathoverflow.net/q/176706.Google Scholar
[355] Willerton, S.. On the magnitude of spheres, surfaces and other homogeneous spaces. Geometriae Dedicata, 168:291310, 2014.Google Scholar
[356] Willerton, S.. The Legendre–Fenchel transform from a category theoretic perspective. Preprint arXiv:1501.03791, available at arXiv.org, 2015.Google Scholar
[357] Willerton, S.. Spread: a measure of the size of metric spaces. International Journal of Computational Geometry and Applications, 25(3):207225, 2015.Google Scholar
[358] Willerton, S.. On the magnitude of odd balls via potential functions. Preprint arXiv:1804.02174, available at arXiv.org, 2018.Google Scholar
[359] Willerton, S.. The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials. Discrete Analysis, 5:142, 2020.Google Scholar
[360] Wilson, M. V. and Shmida, A.. Measuring beta diversity with presence-absence data. Journal of Ecology, 72:10551064, 1984.Google Scholar
[361] Yule, G. U.. The Statistical Study of Literary Vocabulary. Cambridge University Press, Cambridge, 1944.Google Scholar
[362] Zagier, D.. The dilogarithm function. In Cartier, P., Julia, B., Moussa, P., and Vanhove, P., editors, Frontiers in Number Theory, Physics and Geometry II, pages 365. Springer, Berlin, 2007.Google Scholar
[363] Zygmund, A.. Trigonometric Series, volume I. Cambridge University Press, Cambridge, 2nd edition, 1959.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • References
  • Tom Leinster, University of Edinburgh
  • Book: Entropy and Diversity
  • Online publication: 21 April 2021
  • Chapter DOI: https://doi.org/10.1017/9781108963558.017
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • References
  • Tom Leinster, University of Edinburgh
  • Book: Entropy and Diversity
  • Online publication: 21 April 2021
  • Chapter DOI: https://doi.org/10.1017/9781108963558.017
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • Tom Leinster, University of Edinburgh
  • Book: Entropy and Diversity
  • Online publication: 21 April 2021
  • Chapter DOI: https://doi.org/10.1017/9781108963558.017
Available formats
×