scholarly journals Computation of L_⊕ for several cubic Pisot numbers

2007 ◽  
Vol Vol. 9 no. 2 ◽  
Author(s):  
Julien Bernat
Keyword(s):  
Central Symmetry ◽  
Maximal Length ◽  
Pisot Numbers ◽  
International Audience ◽  
Integer Base

International audience In this article, we are dealing with β-numeration, which is a generalization of numeration in a non-integer base. We consider the class of simple Parry numbers such that dβ(1) = 0.k1d-1 kd with d ∈ ℕ, d ≥ 2 and k1 ≥ kd ≥ 1. We prove that these elements define Rauzy fractals that are stable under a central symmetry. We use this result to compute, for several cases of cubic Pisot units, the maximal length among the lengths of the finite β-fractional parts of sums of two β-integers, denoted by L_⊕. In particular, we prove that L_⊕ = 5 in the Tribonacci case.

scholarly journals Computation with No Memory, and Rearrangeable Multicast Networks

2014 ◽  
Vol Vol. 16 no. 1 ◽  
Author(s):  
Emeric Gioan ◽  
Serge Burckel ◽  
Emmanuel Thomé
Keyword(s):  
Multicast Routing ◽  
Close Relation ◽  
Linear Mapping ◽  
Maximal Length ◽  
Arbitrary Mapping ◽  
Matrix Decompositions ◽  
Multicast Networks ◽  
International Audience ◽  
Routing Methods

International audience 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.

scholarly journals The facial weak order in finite Coxeter groups

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Aram Dermenjian ◽  
Christophe Hohlweg ◽  
Vincent Pilaud
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Geometric Interpretation ◽  
Weak Order ◽  
Maximal Length ◽  
Lattice Congruence ◽  
Finite Coxeter Group ◽  
International Audience

International audience We investigate a poset structure that extends the weak order on a finite Coxeter group W to the set of all faces of the permutahedron of W. We call this order the facial weak order. We first provide two alternative characterizations of this poset: a first one, geometric, that generalizes the notion of inversion sets of roots, and a second one, combinatorial, that uses comparisons of the minimal and maximal length representatives of the cosets. These characterizations are then used to show that the facial weak order is in fact a lattice, generalizing a well-known result of A. Bjo ̈rner for the classical weak order. Finally, we show that any lattice congruence of the classical weak order induces a lattice congruence of the facial weak order, and we give a geometric interpretation of its classes.

Conditions equivalent to central symmetry of convex curves

1964 ◽  
Vol 60 (4) ◽  
pp. 779-785 ◽  
Author(s):  
P. C. Hammer ◽  
T. Jefferson Smith
Keyword(s):  
Central Symmetry ◽  
Maximal Length ◽  
Convex Curves ◽  
The Family ◽  
Centrally Symmetric ◽  
Planar Body

In this paper we establish that a convex planar body C is centrally symmetric provided either one of the following conditions hold:(1) Each line halving the circumference of the boundary γ of C is a diametral line.(A diametral line is a line intersecting C in a chord of maximal length in the family of parallel chords.)(2) Each line halving the area of C is a diametral line.

The zeta function of the beta transformation

1994 ◽  
Vol 14 (2) ◽  
pp. 237-266 ◽  
Author(s):  
Leopold Flatto ◽  
Jeffrey C. Lagarias ◽  
Bjorn Poonen
Keyword(s):  
Continuous Function ◽  
Zeta Function ◽  
Symbolic Dynamics ◽  
Counting Function ◽  
Minimum Modulus ◽  
Pisot Numbers ◽  
Salem Numbers ◽  
Integer Base ◽  
Image Position ◽  
Totally Disconnected

AbstractThe β-transformation ƒβ(x) = βx(mod 1), for β > 1, has a symbolic dynamics generalizing radix expansions to an integer base. Two important invariants of ƒβ are the (Artin-Mazur) zeta functionwhere Pk counts the number of fixed points of , and the lap-counting function where Lk counts the number of monotonic pieces of the kth iterate . For β-transformations these functions are related by ζβ(z) = (1 − z)Lβ(z). The function ζβ(z) is meromorphic in the unit disk, is holomorphic in {z: |z| < 1/β}, has a simple pole at z = 1/β, and has no other singularities with |z| = 1/β. Let M(β) denote the minimum modulus of any pole of ζβ(z) in |z| < 1 other than z = 1/β, and set M(β) = 1 if no other pole exists with |z| < 1. Then Pk = βk + O((M(β)−1+ε)k) for any ε > 0. This paper shows that M(β) is a continuous function, that ( for all β, and that An asymptotic formula is derived for M(β) as β → 1+, which implies that M(β) < 1 for all β in an interval (1, 1 + c0). The set is shown to have properties analogous to the set of Pisot numbers. It is closed, totally disconnected, has smallest element ≥ 1 + C0 and contains infinitely many β falling in each interval [n, n + 1) for n ∈ ℤ+. All known members of are algebraic integers which are either Pisot or Salem numbers.

Tandem scanning reflected light microscopy: How it works and applications

1986 ◽  
Vol 44 ◽  
pp. 84-87
Author(s):  
Alan Boyde ◽  
Milan Hadravský ◽  
Mojmír Petran ◽  
Timothy F. Watson ◽  
Sheila J. Jones ◽  
...  
Keyword(s):  
Light Microscopy ◽  
Focal Plane ◽  
Central Symmetry ◽  
Single Line ◽  
Scanning Microscope ◽  
Reflected Light ◽  
Illuminating Light ◽  
Illuminating Beam

The principles of tandem scanning reflected light microscopy and the design of recent instruments are fully described elsewhere and here only briefly. The illuminating light is intercepted by a rotating aperture disc which lies in the intermediate focal plane of a standard LM objective. This device provides an array of separate scanning beams which light up corresponding patches in the plane of focus more intensely than out of focus layers. Reflected light from these patches is imaged on to a matching array of apertures on the opposite side of the same aperture disc and which are scanning in the focal plane of the eyepiece. An arrangement of mirrors converts the central symmetry of the disc into congruency, so that the array of apertures which chop the illuminating beam is identical with the array on the observation side. Thus both illumination and “detection” are scanned in tandem, giving rise to the name Tandem Scanning Microscope (TSM). The apertures are arranged on Archimedean spirals: each opposed pair scans a single line in the image.

Somethin’ about Scandinavia: Danish Shorts on the Post-war International Scene

2018 ◽  
Author(s):  
C. Claire Thomson
Keyword(s):  
United Nations ◽  
International Collaboration ◽  
Marshall Plan ◽  
The Third ◽  
Animated Film ◽  
International Audience ◽  
Post War ◽  
Nordic Cooperation

Building on the picture of post-war Anglo-Danish documentary collaboration established in the previous chapter, this chapter examines three cases of international collaboration in which Dansk Kulturfilm and Ministeriernes Filmudvalg were involved in the late 1940s and 1950s. They Guide You Across (Ingolf Boisen, 1949) was commissioned to showcase Scandinavian cooperation in the realm of aviation (SAS) and was adopted by the newly-established United Nations Film Board. The complexities of this film’s production, funding and distribution are illustrative of the activities of the UN Film Board in its first years of operation. The second case study considers Alle mine Skibe (All My Ships, Theodor Christensen, 1951) as an example of a film commissioned and funded under the auspices of the Marshall Plan. This US initiative sponsored informational films across Europe, emphasising national solutions to post-war reconstruction. The third case study, Bent Barfod’s animated film Noget om Norden (Somethin’ about Scandinavia, 1956) explains Nordic cooperation for an international audience, but ironically exposed some gaps in inter-Nordic collaboration in the realm of film.

Conclusion

2018 ◽  
Author(s):  
Alistair Fox
Keyword(s):  
National Identity ◽  
New Zealand ◽  
Identity Formation ◽  
Essential Role ◽  
Coming Of Age ◽  
Financial Returns ◽  
Self Recognition ◽  
International Audience ◽  
National Identity Formation

The conclusion reaffirms the essential role played by cinema generally, and the coming-of-age genre in particular, in the process of national identity formation, because of its effectiveness in facilitating self-recognition and self-experience through a process of triangulation made possible, for the most part, by a dialogue with some of the nation’s most iconic works of literature. This section concludes by point out the danger posed, however, by an observable trend toward generic standardization in New Zealand films motivated by a desire to appeal to an international audience out of consideration for the financial returns expected by funding bodies under current regimes.

Early Cinema in Scotland

2018 ◽  
Keyword(s):  
Social History ◽  
Small Towns ◽  
Early Cinema ◽  
Popular Song ◽  
Feature Film ◽  
Local Newspapers ◽  
The Social ◽  
Production History ◽  
History Of ◽  
International Audience

This collection of essays, drawn from a three-year AHRC research project, provides a detailed context for the history of early cinema in Scotland from its inception in 1896 till the arrival of sound in the early 1930s. It details the movement from travelling fairground shows to the establishment of permanent cinemas, and from variety and live entertainment to the dominance of the feature film. It addresses the promotion of cinema as a socially ‘useful’ entertainment, and, distinctively, it considers the early development of cinema in small towns as well as in larger cities. Using local newspapers and other archive sources, it details the evolution and the diversity of the social experience of cinema, both for picture goers and for cinema staff. In production, it examines the early attempts to establish a feature film production sector, with a detailed production history of Rob Roy (United Films, 1911), and it records the importance, both for exhibition and for social history, of ‘local topicals’. It considers the popularity of Scotland as an imaginary location for European and American films, drawing their popularity from the international audience for writers such as Walter Scott and J.M. Barrie and the ubiquity of Scottish popular song. The book concludes with a consideration of the arrival of sound in Scittish cinemas. As an afterpiece, it offers an annotated filmography of Scottish-themed feature films from 1896 to 1927, drawing evidence from synopses and reviews in contemporary trade journals.

Specific Features Pertinent to the Use of Planar Antenna Arrays with Central Symmetry in Interference Cancellers

Vestnik MEI ◽  
2018 ◽  
Vol 2 (2) ◽  
pp. 123-128
Author(s):  
Pavel S. Gribov ◽  
◽  
Maria A. Gribova ◽  
Aleksandr Yu. Shatilov ◽  
◽  
...  
Keyword(s):  
Antenna Arrays ◽  
Central Symmetry ◽  
Planar Antenna
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