Handbook of Set Theory, Volume I, Akihiro Kanamori, 0. Introduction Thomas Jech, 1. Stationary Sets Andras Hajnal and Jean Larson, 2. Partition Relations Stevo Todorcevic, 3. Coherent Sequences Greg Hjorth, 4. Borel Equivalence Relations Uri Abraham, 5. Proper Forcing Andreas Blass, 6. Combinatorial Cardinal Characteristics of the Continuum Tomek Bartoszynski, 7. Invariants of Measure and Category Sy Friedman, 8. Constructibility and Class Forcing 48 Ralf-Dieter Schindler and Martin Zeman, 9. Fine Structure 52 Philip Welch, 10. S* Fine Structure 80 Volume II, Patrick Dehornoy, 11. Elementary Embeddings and Algebra James Cummings, 12. Iterated Forcing and Elementary Embeddings Matthew Foreman, 13. Ideals and Generic Elementary Embeddings Uri Abraham and Menachem Magidor, 14. Cardinal Arithmetic Todd Eisworth, 15. Successors of Singular Cardinals Moti Gitik, 16. Prikry-Type Forcings Volume III, William Mitchell, 17. Beginning Inner Model Theory William Mitchell, 18. The Covering Lemma John Steel, 19. An Outline of Inner Model Theory Ernest Schimmerling, 20. A Core Model Tool Box and Guide Steve Jackson, 21. Structural Consequences of AD Itay Neeman, 22. Determinacy in L(R) Peter Koellner and Hugh Woodin, 23. Large Cardinals from Determinacy Paul Larson, 24. Forcing over Models of Determinacy

Handbook of Set Theory

FOUNDATIONS OF GARSIDE THEORYPatrick DEHORNOYwithFranc¸ois DIGNEEddy GODELLEDaan KRAMMERJean MICHELThis text consists of the introduction, table of contents, and bibliography of along manuscript (703 pages) that is currently submitted for publication. This man-uscript develops an extension of Garside’s approach to braid groups and provides auniﬁed treatment for the various algebraic structures that appear in this context.The complete text can be found athttp://www.math.unicaen.fr/∼garside/Garside.pdf.Comments are welcome.IntroductionAnatural,butslowlyemergingprogram. In his PhD thesis prepared under the su-pervision of Graham Higman and defended in 1965 [113], and in the article that fol-lowed [114], Frank A.Garside (1915–1988)solved the Conjugacy Problem of Artin’sbraid group B

Foundations of Garside Theory

A Fast Method for Comparing Braids

Environ dix ans ont passe depuis la decouverte du caractere ordonnable des groupes de tresses, et des methodes diverses ont ete proposees pour expliquer le phenomene. Le but de ce texte est de presenter ces approches variees, qui mettent en jeu l'algebre auto-distributive, les arbres finis, la theorie combinatoire des groupes, les groupes de diffeomorphismes, la theorie des laminations, et la geometrie hyperbolique.

Why are braids orderable

We consider the Artin groups of dihedral type I2(k) defined by the presentation Ak = 〈a,b | prod(a,b;k) = prod(b,a;k)〉 where prod(s,t;k) = ststs …, with k terms in the product on the right-hand side. We prove that the spherical growth series and the geodesic growth series of Ak with respect to the Artin generators {a,b,a-1, b-1} are rational. We provide explicit formulas for the series.

/pdf/growth-series-for-artin-groups-of-dihedral-type-2ks4ui0cqi.pdf

Growth series for Artin groups of dihedral type

The wellordering on positive braids

Closed iterative calculus

We investigate the computation of mappings from a set S^n to itself with "in situ programs", that is using no extra variables than the input, and performing modifications of one component at a time, hence using no extra memory. In this paper, we survey this problem introduced in previous papers by the authors, we detail its close relation with rearrangeable multicast networks, and we provide new results for both viewpoints. A bijective mapping can be computed by 2n-1 component modifications, that is by a program of length 2n-1, a result equivalent to the rearrangeability of the concatenation of two reversed butterfly networks. For a general arbitrary mapping, we give two methods to build a program with maximal length 4n-3. Equivalently, this yields rearrangeable multicast routing methods for the network formed by four successive butterflies with alternating reversions. The first method is available for any set S and practically equivalent to a known method in network theory. The second method, a refinment of the first, described when |S| is a power of 2, is new and allows more flexibility than the known method. For a linear mapping, when S is any field, or a quotient of an Euclidean domain (e.g Z/sZ for any integer s), we build a program with maximal length 2n-1. In this case the assignments are also linear, thereby particularly efficient from the algorithmic viewpoint, and giving moreover directly a program for the inverse when it exists. This yields also a new result on matrix decompositions, and a new result on the multicast properties of two successive reversed butterflies. Results of this flavour were known only for the boolean field Z/2Z.

http://www.lirmm.fr/~gioan/telecharger/2014%20Burckel,%20Gioan,%20Thome%20-%20computation%20with%20no%20memory%20and%20rearrangeable%20multicast%20networks%20-%20DMTCS.pdf

Computation with No Memory, and Rearrangeable Multicast Networks

We investigate the computation of mappings from a set S n to itself with in situ programs , that is using no extra variables than the input, and performing modifications of one component at a time. We consider several types of mappings and obtain effective computation and decomposition methods, together with upper bounds on the program length (number of assignments). Our technique is combinatorial and algebraic (graph coloration, partition ordering, modular arithmetics).

For general mappings, we build a program with maximal length 5n *** 4, or 2n *** 1 for bijective mappings. The length is reducible to 4n *** 3 when |S | is a power of 2. This is the main combinatorial result of the paper, which can be stated equivalently in terms of multistage interconnection networks as: any mapping of {0,1} n can be performed by a routing in a double n -dimensional Benes network. Moreover, the maximal length is 2n *** 1 for linear mappings when S is any field, or a quotient of an Euclidean domain (e.g. ***/s ***). In this case the assignments are also linear, thereby particularly efficient from the algorithmic viewpoint.

The in situ trait of the programs constructed here applies to optimization of program and chip design with respect to the number of variables, since no extra writing memory is used. In a non formal way, our approach is to perform an arbitrary transformation of objects by successive elementary local transformations inside these objects only with respect to their successive states.

/pdf/mapping-computation-with-no-memory-4vrm3h7wcw.pdf

Mapping Computation with No Memory

We construct, for each integer n,
 three functions from {0,1} n to {0,1} such that any boolean mapping from
 {0,1} n to {0,1} n can be computed with a finite sequence of assignations
 only using the n input variables and those three functions.

/pdf/three-generators-for-minimal-writing-space-computations-4zy66tavsr.pdf

Serge Burckel

Papers

The wellordering on positive braids

Closed iterative calculus

Computation with No Memory, and Rearrangeable Multicast Networks

Mapping Computation with No Memory

Three generators for minimal writing-space computations