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An axiomatic approach to algebrization

Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova
2009 Proceedings of the 41st annual ACM symposium on Symposium on theory of computing - STOC '09  
In this paper, building on the work of Arora, Impagliazzo, and Vazirani [4], we propose an axiomatic approach to "algebrization", which complements and clarifies the approaches of [12] and [1].  ...  to one oracle to classes relative to an algebraic extension of that oracle.  ...  We want to thank Scott Aaronson, Lance Fortnow, and the anonymous referees for their comments.  ... 
doi:10.1145/1536414.1536509 dblp:conf/stoc/ImpagliazzoKK09 fatcat:o6tthbtumnde3criihmkxnzxbm

Page 924 of Mathematical Reviews Vol. 37, Issue 4 [page]

1969 Mathematical Reviews  
Ludwig, Giinther 5003 Attempt of an axiomatic foundation of quantum mechanics and more general theories. II. Comm. Math. Phys. 4 (1967), 331-348.  ...  Jauch (Geneva) Ludwig, Giinther 5004 Attempt of an axiomatic foundation of quantum mechanics and more general theories. III. Comm. Math. Phys. 9 (1968), 1-12.  ... 

Page 516 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 56, Issue 6 [page]

1950 American Mathematical Society. Bulletin of the American Mathematical Society  
Steenrod, Axiomatic approach to homology theory, Proc. Nat. Acad. Sci. U. S. A. vol. 31 (1945) pp. 117-120. 10. , Foundations of algebraic topology, to appear. 11. K.  ...  516 SAUNDERS MacLANE tions between vector spaces) can be deduced from our self-dual sys- tem of axioms for an abelian bicategory.  ... 

Page 3145 of Mathematical Reviews Vol. , Issue 2004d [page]

2004 Mathematical Reviews  
From the introduction: “The purpose of the present note is to show how the axiomatic approach to Tate cohomology of J. Greenlees [J. Pure Appl.  ...  In an erratum [Algebr. Geom. Topol., submitted] the author has corrected several mistakes and misleading statements.  ... 

Page 9337 of Mathematical Reviews Vol. , Issue 2002M [page]

2002 Mathematical Reviews  
The author advocates an innovative approach to quantized gauge theories using mathematically well-defined ob- jects developed within the framework of axiomatic quantum field theory and causal perturbation  ...  Within this approach all couplings are determined by perturbative gauge invariance without the use of spontaneous symmetry breaking. As an application the elec- troweak theory is constructed.  ... 

Page 4969 of Mathematical Reviews Vol. , Issue 84m [page]

1984 Mathematical Reviews  
Then a unifying approach for the Brelot and Hervé decomposition is presented.  ...  Finally an abstract decomposition result is proved within the framework of potential cones showing the validity of Brelot’s decomposition in more general axiomatics.  ... 

Algèbre noethérienne non commutative, by L. Lesieur et R. Croisot. [Mém. Sci. Math. 154] Gauthier-Villars, Paris 1963. 117 pages

B. H. Neumann
1964 Canadian mathematical bulletin  
One of the reasons for the reluctance of senior faculty members to teach the course is the lack of discipline and organization in the standard textbook.  ...  Any mathematician will find it a genuine pleasure to teach from this reprinting of Professor Albert 1 s book. In the preface the author states: "College Algebra has a basic unity.  ...  To achieve this object, he approaches the subject from a semi-axiomatic point of view and uses a style which is always explicit rather than concise.  ... 
doi:10.1017/s0008439500031957 fatcat:mm5ejwgjobbufcvbvulecypt2q

Page 1300 of Mathematical Reviews Vol. , Issue 80D [page]

1980 Mathematical Reviews  
The author gives an axiomatization of Birkhoff and von Neumann’s quantum logic extended by the definition of the func- tor —,, i=0,1,2,3,4, an axiomatization of E(M.,, ;) and axiomatiza- tions of E(M,,  ...  This paper contains a short elementary introduction to nonstan- dard analysis very similar to the classical approach based on ultra- powers.  ... 

Page 7720 of Mathematical Reviews Vol. , Issue 2004j [page]

2004 Mathematical Reviews  
Level 0 of such an axiomatization consists in requiring the tensor product to be exact in each variable. This is what everybody agrees about and this is already a workable framework.  ...  (English summary) Algebr. Represent. Theory 6 (2003), no. 5, 475-499.  ... 

Page 1807 of Mathematical Reviews Vol. 50, Issue 6 [page]

1975 Mathematical Reviews  
The author chooses an axiomatic approach to category theory without using any set theory: the usual axioms describing the inner structures of categories, the axioms for functors together with the axiom  ...  In addition to the articles cited above a similar point of view was expressed in an article by M. F. Atiyah [Bild Wissensch. 6 (1969), 314-323]. J. S. Joel (Ann Arbor, Mich.)  ... 

Page 580 of Mathematical Reviews Vol. 50, Issue 3 [page]

1975 Mathematical Reviews  
When it first appeared it was one of the few books that presented a modern approach to mathematics (including an axiomatic approach to algebra) at a really elementary level and with the clarity that that  ...  E. 4176 ntroduction a |’algébre moderne.  ... 

Les fondements de la géométrie. Tome II: Géométrie projective par Bêla Kerékjarto. Gauthier-Villars, Paris, et Académie des Sciences hongroise, 1966. 528 pages. $14.50

I. M. Yaglom
1968 Canadian mathematical bulletin  
élémentaire", Paris, Hermann 1964, as well as to the more recently developed approaches to euclidean geometry (cf. e.g.  ...  Today, however, when everywhere less and less attention is given to Hilbert's axiomatical scheme, an exposition of projective geometry on this basis will hardly carry along the imagination of young geometers  ... 
doi:10.1017/s0008439500028988 fatcat:2atswe5azrcojl53ms7r2ryo4i

Page 3542 of Mathematical Reviews Vol. , Issue 94f [page]

1994 Mathematical Reviews  
It presents an interesting discussion on the axiomatization of quantum mechanics.  ...  Differ- ent approaches are compared, in particular, the traditional setting where observables are represented by selfadjoint operators on a Hilbert space and the axiomatization where observables are rep  ... 

Page 3271 of Mathematical Reviews Vol. , Issue 85h [page]

1985 Mathematical Reviews  
His semantic approach led to the universal algebraic concept of a relation algebra.  ...  to an operation free sentence in at most three variables.  ... 

Page 2007 of Mathematical Reviews Vol. 56, Issue 6 [page]

1978 Mathematical Reviews  
The author treats in a very elementary way sore of the ideas that have been used in the development of approaches to Hilbert’s first problem: the use of model theory and the work of P. J.  ...  The article ends with an interpretation of what it would mean for X-automata to have C above as a groupoid, i.e., to have Booleanness, hence classical logic.  ... 
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