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Books on the topic 'Fundamental theorem of arithmetic'

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Published: 4 June 2021
Last updated: 4 February 2022

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1

service), SpringerLink (Online, ed. The Arithmetic of Fundamental Groups: PIA 2010. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012.

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2

Stix, Jakob. Rational Points and Arithmetic of Fundamental Groups: Evidence for the Section Conjecture. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013.

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3

Benjamin, Fine. The fundamental theorem of algebra. New York: Springer, 1997.

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4

Fine, Benjamin, and Gerhard Rosenberger. The Fundamental Theorem of Algebra. New York, NY: Springer New York, 1997. http://dx.doi.org/10.1007/978-1-4612-1928-6.

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5

Aitken, Wayne. An arithmetic Riemann-Roch theorem for singular arithmetic surfaces. Providence, R.I: American Mathematical Society, 1996.

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6

Faltings, Gerd. Lectures on the arithmetic Riemann-Roch theorem. Princeton, N.J: Princeton University Press, 1992.

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7

Frege's theorem. Oxford: Clarendon Press, 2011.

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8

National Institute of Public Finance and Policy (India), ed. The second fundamental theorem of positive economics. New Delhi: National Institute of Public Finance and Policy, 2012.

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9

Stix, Jakob, ed. The Arithmetic of Fundamental Groups. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-23905-2.

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10

Kyōto Daigaku. Sūri Kaiseki Kenkyūjo. Communications in arithmetic fundamental groups. [Kyoto]: Kyōto Daigaku Sūri Kaiseki Kenkyūjo, 2002.

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11

Fried, Michael, and Yasutaka Ihara, eds. Arithmetic Fundamental Groups and Noncommutative Algebra. Providence, Rhode Island: American Mathematical Society, 2002. http://dx.doi.org/10.1090/pspum/070.

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12

Quaife, Art. Automated development of fundamental mathematical theories. Dordrecht: Kluwer Academic, 1992.

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13

Fermat's last theorem. Providence, Rhode Island: American Mathematical Society, 2013.

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14

Stix, Jakob. Rational Points and Arithmetic of Fundamental Groups. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-30674-7.

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15

Benjamin, Fine. To themeliōdes theōrēma tēs algevras. Athēna: Ekdoseis Leader Books, 2001.

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16

The optimal version of Hua's fundamental theorem of geometry of rectangular matrices. Providence, Rhode Island: American Mathematical Society, 2014.

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17

Bardy, Nicole. Systèmes de racines infinis. [Paris, France]: Société mathématique de France, 1996.

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18

Capacity theory with local rationality: The strong Fekete-Szegő theorem on curves. Providence, Rhode Island: American Mathematical Society, 2013.

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19

1942-, Fried Michael D., and Ihara Y. 1938-, eds. Arithmetic fundamental groups and noncommutative algebra: 1999 Von Neumann Conference on Arithmetic Fundamental Groups and Noncommutative Algebra, August 16-27, 1999, Mathematical Sciences Research Institute, Berkeley, California. Providence, R.I: American Mathematical Society, 2002.

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20

Doran, Edward. Fundamental mathematics for college and technical students: A study of arithmetic, calculator, algebra, geometry, trigonometry. 2nd ed. Broomfield, CO: Finesse Pub. Co., 1988.

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21

Thomas, Taylor. Ē theōrētikē arithmētikē tōn pythagoreiōn. Athēna: Iamvlichos, 1995.

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22

Stix, Jakob. The Arithmetic of Fundamental Groups: PIA 2010. Springer, 2012.

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23

The Arithmetic of Fundamental Groups Contributions in Mathematical and Computational Sciences. Springer, 2012.

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24

Rational Points and Arithmetic of Fundamental Groups Lecture Notes in Mathematics. Springer, 2012.

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25

Earl, Richard, and James Nicholson. The Concise Oxford Dictionary of Mathematics. 6th ed. Oxford University Press, 2021. http://dx.doi.org/10.1093/acref/9780198845355.001.0001.

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Over 4,000 entries This informative A to Z provides clear, jargon-free definitions of a wide variety of mathematical terms. Its articles cover both pure and applied mathematics and statistics, and include key theories, concepts, methods, programmes, people, and terminology. For this sixth edition, around 800 new terms have been defined, expanding on the dictionary’s coverage of algebra, differential geometry, algebraic geometry, representation theory, and statistics. Among this new material are articles such as cardinal arithmetic, first fundamental form, Lagrange’s theorem, Navier-Stokes equations, potential, and splitting field. The existing entries have also been revised and updated to account for developments in the field. Numerous supplementary features complement the text, including detailed appendices on basic algebra, areas and volumes, trigonometric formulae, and Roman numerals. Newly added to these sections is a historical timeline of significant mathematicians’ lives and the emergence of key theorems. There are also illustrations, graphs, and charts throughout the text, as well as useful web links to provide access to further reading.
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26

Okasha, Samir. Wright’s Adaptive Landscape, Fisher’s Fundamental Theorem. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198815082.003.0004.

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Fitness maximization, or optimization, is a controversial idea in evolutionary biology. One classical formulation of this idea is that natural selection will tend to push a population up a peak in an adaptive landscape, as Sewall Wright first proposed. However, the hill-climbing property only obtains under particular conditions, and even then the ascent is not usually by the steepest route; this shows why it is misleading to assimilate the process of natural selection to a process of goal-directed choice. A different formulation of the idea of fitness-maximization is R. A. Fisher’s ‘fundamental theorem of natural selection’. However, the theorem points only to a weak sense in which selection is an optimizing process, for it requires that ‘environmental constancy’ be understood in a highly specific way. It does not vindicate the claim that natural selection has an intrinsic tendency to produce adaptation.
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27

McLarty, Colin. The Roles of Set Theories in Mathematics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198748991.003.0001.

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What mathematicians know and use about sets varies across branches of mathematics but rarely includes such fundamental aspects of Zermelo–Fraenkel (ZF) set theory as the iterative hierarchy. All mathematicians know and use the axioms of the Elementary Theory of the Category of Sets (ETCS), though few know ETCS or any set theory by name. The chapter depicts the iterative hierarchy of ZF and constructibility as gauge theories. Since gauge theories are prominently used in physics, so these are used in work on the continuum hypothesis, large cardinals, and provability in arithmetic. But mathematicians outside logic avoid these gauges and work with structures only up to isomorphism, as does ETCS.
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28

Heck, Richard G. Frege's Theorem. Oxford University Press, 2014.

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29

Von Neumann Conference on Arithmetic Fundamental Groups and noncommuta. Arithmetic Fundamental Groups and Noncommutative Algebra. American Mathematical Society, 2002.

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30

Talbott, George Robert. Derivatives in Cardinal Arithmetic. Lotus Press, 2002.

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31

Lee, Kuen Hung. Fundamental Arithmetic: A Step-By-Step Approach. 2nd ed. Edmund Pub Co, 2002.

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32

Faltings, Gerd. Lectures on the Arithmetic Riemann-Roch Theorem. (AM-127), Volume 127. Princeton University Press, 2016.

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33

E, Wells David, and University of New Brunswick. Department of Surveying Engineering., eds. GPS design: Undifferenced carrier beat phase observations and the fundamental differencing theorem. Fredericton, N.B: Dept. of Surveying Engineering, University of New Brunswick, 1987.

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34

Farb, Benson, and Dan Margalit. The Dehn-Nielsen-Baer Theorem. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691147949.003.0009.

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This chapter deals with the Dehn–Nielsen–Baer theorem, one of the most beautiful connections between topology and algebra in the mapping class group. It begins by defining the objects in the statement of the Dehn–Nielsen–Baer theorem, including the extended mapping class group and outer automorphism groups. It then considers the use of the notion of quasi-isometry in Dehn's original proof of the Dehn–Nielsen–Baer theorem. In particular, it discusses a theorem on the fundamental observation of geometric group theory, along with the property of being linked at infinity. It also presents the proof of the Dehn–Nielsen–Baer theorem and an analysis of the induced homeomorphism at infinity before concluding with two other proofs of the Dehn–Nielsen–Baer theorem, one inspired by 3-manifold theory and one using harmonic maps.
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35

Bryant, Nerissa Bell. Mathematics in Daily Living: Fundamental Algebra (Mathematics in Daily Living). Steck-Vaughn, 1985.

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36

Grand Unified Theorem: Discovery of the Theory of Everything and the Fundamental Building Block of Quantum Theory. Nova Science Publishers, 2004.

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37

(Editor), S. G. Dani, and Gopal Prasad (Editor), eds. Algebraic Groups and Arithmetic (Tata Institute of Fundamental Research, Studies in Mathematics, No. 17). Narosa Publishing House, 2004.

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38

Wüstholz, Gisbert, and Clemens Fuchs, eds. Arithmetic and Geometry. Princeton University Press, 2019. http://dx.doi.org/10.23943/princeton/9780691193779.001.0001.

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This book presents highlights of recent work in arithmetic algebraic geometry by some of the world's leading mathematicians. Together, these 2016 lectures—which were delivered in celebration of the tenth anniversary of the annual summer workshops in Alpbach, Austria—provide an introduction to high-level research on three topics: Shimura varieties, hyperelliptic continued fractions and generalized Jacobians, and Faltings heights and L-functions. The book consists of notes, written by young researchers, on three sets of lectures or minicourses given at Alpbach. The first course contains recent results dealing with the local Langlands conjecture. The fundamental question is whether for a given datum there exists a so-called local Shimura variety. In some cases, they exist in the category of rigid analytic spaces; in others, one has to use Scholze's perfectoid spaces. The second course addresses the famous Pell equation—not in the classical setting but rather with the so-called polynomial Pell equation, where the integers are replaced by polynomials in one variable with complex coefficients, which leads to the study of hyperelliptic continued fractions and generalized Jacobians. The third course originates in the Chowla–Selberg formula and relates values of the L-function for elliptic curves with the height of Heegner points on the curves. It proves the Gross–Zagier formula on Shimura curves and verifies the Colmez conjecture on average.
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39

Algebra, Arithmetic and Geometry Mumbai 2000 (Parts I and II) (Tata Institute of Fundamental Research, Bombay// Studies in Mathematics). Narosa Pub House, 2002.

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40

Abbes, Ahmed, and Michel Gros. Representations of the fundamental group and the torsor of deformations. Local study. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691170282.003.0002.

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This chapter focuses on representations of the fundamental group and the torsor of deformations. It considers the case of an affine scheme of a particular type, qualified also as small by Faltings. It introduces the notion of Dolbeault generalized representation and the companion notion of solvable Higgs module, and then constructs a natural equivalence between these two categories. It proves that this approach generalizes simultaneously Faltings' construction for small generalized representations and Hyodo's theory of p-adic variations of Hodge–Tate structures. The discussion covers the relevant notation and conventions, results on continuous cohomology of profinite groups, objects with group actions, logarithmic geometry lexicon, Faltings' almost purity theorem, Faltings extension, Galois cohomology, Fontaine p-adic infinitesimal thickenings, Higgs–Tate torsors and algebras, Dolbeault representations, and small representations. The chapter also describes the descent of small representations and applications and concludes with an analysis of Hodge–Tate representations.
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41

Venkataramana, T. N., and International Conference on Cohomology o. Proceedings of the International Conference on Cohomology of Arithmetic Groups, L-Functions and Automorphic Forms, Mumbai 1998 (Tata Institute of Fundamental Research, Bombay// Studies in Mathematics). Alpha Science International, Ltd, 2002.

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42

Brechenmacher, Frédéric. Algebraic generality versus arithmetic generality in the 1874 controversy between C. Jordan and L. Kronecker. Edited by Karine Chemla, Renaud Chorlay, and David Rabouin. Oxford University Press, 2017. http://dx.doi.org/10.1093/oxfordhb/9780198777267.013.16.

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This article revisits the 1874 controversy between Camille Jordan and Leopold Kronecker over two theorems, namely Jordan’s canonical forms and Karl Weierstrass’s elementary divisors theorem. In particular, it compares the perspectives of Jordan and Kronecker on generality and how their debate turned into an opposition over the algebraic or arithmetic nature of the ‘theory of forms’. It also examines the ways in which the various actors used the the categories of algebraic generality and arithmetic generality. After providing a background on the Jordan-Kronecker controversy, the article explains Jordan’s canonical reduction and Kronecker’s invariant computations in greater detail. It argues that Jordan and Kronecker aimed to ground the ‘theory of forms’ on new forms of generality, but could not agree on the types of generality and on the treatments of the general they were advocating.
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43

Weekly Reader Early Learning Library (Firm), ed. I know same and different =: Igual y diferente. Milwaukee, WI: Weekly Reader Early Learning Library, 2006.

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44

Nicholson, Peter. The Principles of Architecture, Containing the Fundamental Rules of the Art, in Geometry, Arithmetic, and Mensuration, with the Application of Those Rules to Practice: Volume 1. Adamant Media Corporation, 2001.

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45

Nicholson, Peter. The Principles of Architecture, Containing the Fundamental Rules of the Art, in Geometry, Arithmetic, and Mensuration, with the Application of Those Rules to Practice: Volume 2. Adamant Media Corporation, 2001.

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46

McDuff, Dusa, and Dietmar Salamon. Symplectic manifolds. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198794899.003.0004.

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The third chapter introduces the basic notions of symplectic topology, such as symplectic forms, symplectomorphisms, and Lagrangian submanifolds. A fundamental classical construction is Moser isotopy, with its various applications such as Darboux’s theorem and the Lagrangian neighbourhood theorem. The chapter now includes a brief discussion of the Chekanov torus and Luttinger surgery. The last section on contact structures has been significantly expanded.
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47

Walsh, Bruce, and Michael Lynch. Theorems of Natural Selection: Results of Price, Fisher, and Robertson. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198830870.003.0006.

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This chapter reviews a number of “theorems” of natural selection. These include exact results (true mathematical theorems): the Robertson-Price identity, Price's general expression for any form of selection response, and the Fisher-Price-Ewens version of Fisher's fundamental theorem. Their generality comes as the cost of usually being very difficult to apply. An important exception is the Robertson-Price identity, which expresses the within-generation change in the mean of a trait as its covariance with relative fitness. This chapter also examines three classic approximations: Fisher's fundamental theorem for the behavior of mean population fitness, and Robertson's secondary theorem and the breeder's equation for the expected response in a trait under selection, showing both how these results are connected and the error given by the various approximations. Finally, the chapter examines the connection between the additive variance of a trait and its correlation with fitness.
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48

Button, Tim, and Sean Walsh. Internal categoricity and the natural numbers. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198790396.003.0010.

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The simple conclusion of the preceding chapters is that moderate modelism fails. But this leaves us with a choice between abandoning moderation and abandoning modelism. The aim of this chapter, and the next couple of chapters, is to outline a speculative way to save moderation by abandoning modelism. The idea is to do metamathematics without semantics, by working deductively in a higher-order logic. In this chapter, the focus is on the internal categoricity of arithmetic. After formalising an internal notion of a model of the Peano axioms, we show how to internalise Dedekind’s Categority Theorem. The resulting “intolerance” of Peano arithmetic provides internalists with a way to draw the distinction between algebraic and univocal theories. In the appendices, we discuss how this relates to Parsons’ important work, and establish a certain dependence of the internal categoricity theorem on higher-order logic.
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49

Button, Tim, and Sean Walsh. Internal categoricity and the sets. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198790396.003.0011.

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As the previous chapter discussed the internalist perspective on the categoricity of arithmetic, this chapter presents the internalist perspective on sets. In particular, we show both how to internalise Scott-Potter set theory its quasi-categoricity theorem, and how to internalise Zermelo’s Quasi-Categoricity Theorem. As in the case of arithmetic, this gives a non-semantic way to draw the boundary between algebraic and univocal theories. A particularly compelling case of the quasi-univocity of set theory revolves around the continuum hypothesis. Furthermore, by additionally postulating that the size of the pure sets is the same as the size of the universe, these famous quasi-categoricity results can actually be turned into internal categoricity results simpliciter, so that one has full univocity instead of mere quasi-univocity. In the appendices we prove these results, and we discuss how they relate to important work by McGee and Martin.
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50

Button, Tim, and Sean Walsh. Categoricity and the natural numbers. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198790396.003.0007.

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This chapter focuses on modelists who want to pin down the isomorphism type of the natural numbers. This aim immediately runs into two technical barriers: the Compactness Theorem and the Löwenheim-Skolem Theorem (the latter is proven in the appendix to this chapter). These results show that no first-order theory with an infinite model can be categorical; all such theories have non-standard models. Other logics, such as second-order logic with its full semantics, are not so expressively limited. Indeed, Dedekind's Categoricity Theorem tells us that all full models of the Peano axioms are isomorphic. However, it is a subtle philosophical question, whether one is entitled to invoke the full semantics for second-order logic — there are at least four distinct attitudes which one can adopt to these categoricity result — but moderate modelists are unable to invoke the full semantics, or indeed any other logic with a categorical theory of arithmetic.
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