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Books on the topic 'Geometry Axioms'

Author: Grafiati
Published: 4 June 2021
Last updated: 1 February 2022

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1

Whitehead, Alfred North. Axioms of projective geometry. [Place of publication not identified]: Nabu Press, 2010.

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2

Xiao-shan, Gao, and Chang Ching-chung 1936-, eds. Machine proofs in geometry: Automated production of readable proofs for geometry theorems. Singapore: World Scientific, 1994.

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3

Goetsch, James Robert. Vico's axioms: The geometry of the human world. New Haven: Yale University Press, 1995.

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4

Fetisov, A. I. Proof in geometry. Mineola, N.Y: Dover Publications, 2006.

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5

Zahlbereiche, naive Mengenlehre, Axiomatik der Geometrie: Grundlagenfragen. Bremen: Universitätsdruckerei Bremen, 2007.

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6

Erro, Luis Enrique. Axioma: El pensamiento matemático contemporáneo. México: Dirección de Difusión Cultural, Departamento Editorial, 1985.

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7

Borceux, Francis. An Axiomatic Approach to Geometry: Geometric Trilogy I. Springer, 2016.

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8

Whitehead, Alfred North. The Axioms of Descriptive Geometry. BiblioBazaar, 2009.

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9

Whitehead, Alfred North. The Axioms Of Descriptive Geometry. Kessinger Publishing, LLC, 2007.

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10

The Axioms of Descriptive Geometry. Dover Publications, 2004.

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11

Series, Michigan Historical Reprint. The axioms of descriptive geometry, by A.N. Whitehead. Scholarly Publishing Office, University of Michigan Library, 2005.

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12

Fetisov, A. I., and Ya S. Dubnov. Proof in Geometry: With "Mistakes in Geometric Proofs". Dover Publications, 2006.

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13

Lee, John M. Axiomatic Geometry (Pure and Applied Undergraduate Texts) (Sally: Pure and Applied Undergraduate Texts). American Mathematical Society, 2013.

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14

Mann, Peter. Calculus of Variations. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0036.

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This chapter presents an introduction to linear algebra. Classical mechanics is best understood in the language of differential geometry, which itself requires a working knowledge of the key concepts in linear algebra. This chapter walks through the required knowledge from this broad discipline and guides the reader towards the goal of the next chapter, differential geometry. Topics discussed include vector spaces, linear maps, basis sets, cobases, inner products, tensors, wedge products and exterior algebra, as well as the axioms of vector space geometry. The chapter concludes with a brief discussion of Grassmann variables, which tend to crop up when classical fermionic fields are defined.
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15

Stillwell, John. Reverse Mathematics. Princeton University Press, 2019. http://dx.doi.org/10.23943/princeton/9780691196411.001.0001.

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Reverse mathematics is a new field that seeks to find the axioms needed to prove given theorems. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentieth-century field of set theory. This book offers a historical and representative view, emphasizing basic analysis and giving a novel approach to logic. It concludes that mathematics is an arena where theorems cannot always be proved outright, but in which all of their logical equivalents can be found. This creates the possibility of reverse mathematics, where one seeks equivalents that are suitable as axioms. By using a minimum of mathematical logic in a well-motivated way, the book will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics.
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16

Stillwell, John. Statistical Inference via Convex Optimization. Princeton University Press, 2019. http://dx.doi.org/10.23943/princeton/9780691197296.001.0001.

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Reverse mathematics is a new field that seeks to find the axioms needed to prove given theorems. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentieth-century field of set theory. This book offers a historical and representative view, emphasizing basic analysis and giving a novel approach to logic. It concludes that mathematics is an arena where theorems cannot always be proved outright, but in which all of their logical equivalents can be found. This creates the possibility of reverse mathematics, where one seeks equivalents that are suitable as axioms. By using a minimum of mathematical logic in a well-motivated way, the book will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics.
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17

Hellman, Geoffrey, and Stewart Shapiro. Non-Euclidean Extensions. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198712749.003.0006.

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This chapter adapts the foregoing results to present two non-Euclidean theories, both in line with the (semi-)Aristotelian theme of rejecting points, as parts of regions (but working with actual infinity). The first theory is a two-dimensional hyperbolic space, that is, one that has a negative constant curvature. The second theory captures a space of constant positive curvature, a two-dimensional spherical geometry. The task here is to formulate axioms on regions which allow us to prove that (i) there are no infinitesimal regions and (ii) that there are no parallels to any given “line” through any “point” not on the given “line”.
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18

Erro, Luis Enrique. Axioma: El pensamiento matematico contemporaneo. Direccion de Difusion Cultural, Departamento Editorial, 1985.

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19

Iliopoulos, John. Symmetries. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198805175.003.0003.

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The concept of symmetry plays a central role in our understanding of the fundamental laws of Nature. Through a deep mathematical theorem due to A.E. Noether, all conservation laws of classical physics are related to symmetries. In this chapter we start from the intuitively obvious notions of translation and rotation symmetries which are part of the axioms of Euclidian geometry. Following W. Heisenberg, we introduce the idea of isospin as a first example of an internal symmetry. A further abstraction leads to the concept of a global versus local, or gauge symmetry, which is a fundamental property of General Relativity. Combining the notions of internal and gauge symmetries we obtain the Yang-Mills theory which describes all fundamental interactions among elementary particles. A more technical part, which relates a gauge symmetry of the Schrödinger equation of quantum mechanics to the electromagnetic interactions, is presented in a separate section and its understanding is not required for the rest of the book.
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20

Ryan, Matthew. The Celebrated Theory Of Parallels: Demonstration Of The Celebrated Theorem; Euclid I, Axiom 12. Kessinger Publishing, LLC, 2007.

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21

Lewes, George Henry. Problems of Life and Mind : The Principles of Certitude. from the Known to the Unknown. Matter and Force. Force and Cause. the Absolute in the Correlations of Feeling and Motion. Appendix : Imaginary Geometry and the Truth of Axioms. Lagrange and Hegel: The. HardPress, 2020.

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22

Awodey, Steve. Structuralism, Invariance, and Univalence. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198748991.003.0004.

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The recent discovery of an interpretation of constructive type theory into abstract homotopy theory suggests a new approach to the foundations of mathematics with intrinsic geometric content and a computational implementation. Voevodsky has proposed such a program, including a new axiom with both geometric and logical significance: the univalence axiom. It captures the familiar aspect of informal mathematical practice according to which one can identify isomorphic objects. This powerful addition to homotopy type theory gives the new system of foundations a distinctly structural character.
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23

Caramello, Olivia. A duality theorem. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198758914.003.0005.

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This chapter presents a duality theorem providing, for each geometric theory, a natural bijection between its geometric theory extensions (also called ‘quotients’) and the subtoposes of its classifying topos. Two different proofs of this theorem are provided, one relying on the theory of classifying toposes and the other, of purely syntactic nature, based on a proof-theoretic interpretation of the notion of Grothendieck topology. Via this interpretation the theorem can be reformulated as a proof-theoretic equivalence between the classical system of geometric logic over a given geometric theory and a suitable proof system whose rules correspond to the axioms defining the notion of Grothendieck topology. The role of this duality as a means for shedding light on axiomatization problems for geometric theories is thoroughly discussed, and a deduction theorem for geometric logic is derived from it.
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24

Baulieu, Laurent, John Iliopoulos, and Roland Sénéor. From Classical to Quantum Fields. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198788393.001.0001.

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Quantum field theory has become the universal language of most modern theoretical physics. This book is meant to provide an introduction to this subject with particular emphasis on the physics of the fundamental interactions and elementary particles. It is addressed to advanced undergraduate, or beginning graduate, students, who have majored in physics or mathematics. The ambition is to show how these two disciplines, through their mutual interactions over the past hundred years, have enriched themselves and have both shaped our understanding of the fundamental laws of nature. The subject of this book, the transition from a classical field theory to the corresponding Quantum Field Theory through the use of Feynman’s functional integral, perfectly exemplifies this connection. It is shown how some fundamental physical principles, such as relativistic invariance, locality of the interactions, causality and positivity of the energy, form the basic elements of a modern physical theory. The standard theory of the fundamental forces is a perfect example of this connection. Based on some abstract concepts, such as group theory, gauge symmetries, and differential geometry, it provides for a detailed model whose agreement with experiment has been spectacular. The book starts with a brief description of the field theory axioms and explains the principles of gauge invariance and spontaneous symmetry breaking. It develops the techniques of perturbation theory and renormalisation with some specific examples. The last Chapters contain a presentation of the standard model and its experimental successes, as well as the attempts to go beyond with a discussion of grand unified theories and supersymmetry.
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25

David Hilbert and the Axiomatization of Physics (1898-1918): From Grundlagen der Geometrie to Grundlagen der Physik (Archimedes). Springer, 2004.

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26

Two temperaments seen through Strindberg's Miss Julie. Lund, Sweden: Copenhagen University, Denmark, 2012.

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27

Huffaker, Ray, Marco Bittelli, and Rodolfo Rosa. Phase Space Reconstruction. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198782933.003.0003.

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In this chapter we introduce an important concept concerning the study of both discrete and continuous dynamical systems, the concept of phase space or “state space”. It is an abstract mathematical construction with important applications in statistical mechanics, to represent the time evolution of a dynamical system in geometric shape. This space has as many dimensions as the number of variables needed to define the instantaneous state of the system. For instance, the state of a material point moving on a straight line is defined by its position and velocity at each instant, so that the phase space for this system is a plane in which one axis is the position and the other one the velocity. In this case, the phase space is also called “phase plane”. It is later applied in many chapters of the book.
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28

Eynard, Bertrand. Random matrices and loop equations. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797319.003.0007.

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This chapter is an introduction to algebraic methods in random matrix theory (RMT). In the first section, the random matrix ensembles are introduced and it is shown that going beyond the usual Wigner ensembles can be very useful, in particular by allowing eigenvalues to lie on some paths in the complex plane rather than on the real axis. As a detailed example, the Plancherel model is considered from the point of RMT. The second section is devoted to the saddle-point approximation, also called the Coulomb gas method. This leads to a system of algebraic equations, the solution of which leads to an algebraic curve called the ‘spectral curve’ which determines the large N expansion of all observables in a geometric way. Finally, the third section introduces the ‘loop equations’ (i.e., Schwinger–Dyson equations associated with matrix models), which can be solved recursively (i.e., order by order in a semi-classical expansion) by a universal recursion: the ‘topological recursion’.
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