scholarly journals Minkowski decompositions of associahedra

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Carsten Lange
Keyword(s):  
Standard Simplex ◽  
Right Hand ◽  
Möbius Inversion ◽  
Minkowski Sums ◽  
International Audience

International audience Realisations of associahedra can be obtained from the classical permutahedron by removing some of its facets and the set of facets is determined by the diagonals of certain labeled convex planar $n$-gons as shown by Hohlweg and Lange (2007). Ardila, Benedetti, and Doker (2010) expressed polytopes of this type as Minkowski sums and differences of scaled faces of a standard simplex and computed the corresponding coefficients $y_I$ by Möbius inversion from the $z_I$ if tight right-hand sides $z_I$ for all inequalities of the permutahedron are assumed. Given an associahedron of Hohlweg and Lange, we first characterise all tight values $z_I$ in terms of non-crossing diagonals of the associated labeled $n$-gon, simplify the formula of Ardila et al., and characterise the remaining terms combinatorially. Dans un article paru en 2007, Hohlweg et Lange décrivent des associaèdres réalisés à partir du permutoèdre en enlevant certaines de ses facettes. Ces facettes sont déterminées par les diagonales d'une famille de $n$-gones étiquetés. En 2010, Ardila, Benedetti et Doker ont montré que ces polytopes s'expriment par des sommes et différences de Minkowski de faces pondérées d'un simplexe. De plus, si les coefficients $z_I$ des inégalités décrivant l'associaèdre à partir du permutoèdre sont optimaux, alors les coefficients $y_I$ correspondants sont calculés par une inversion de Möbius. Étant donné un tel associaèdre, nous décrivons d'abord les valeurs optimales $z_I$ en termes de diagonales non croisées d'un certain $n$-gone étiqueté, ensuite nous simplifions la formule de Ardila et al. pour finalement décrire combinatoirement les termes restants.

scholarly journals New Hopf Structures on Binary Trees

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Stefan Forcey ◽  
Aaron Lauve ◽  
Frank Sottile
Keyword(s):  
Hopf Algebras ◽  
Homotopy Theory ◽  
Algebraic Combinatorics ◽  
Binary Trees ◽  
Independent Interest ◽  
Hopf Module ◽  
Möbius Inversion ◽  
International Audience ◽  
Higher Categories ◽  
Donne Lieu

International audience The multiplihedra $\mathcal{M}_{\bullet} = (\mathcal{M}_n)_{n \geq 1}$ form a family of polytopes originating in the study of higher categories and homotopy theory. While the multiplihedra may be unfamiliar to the algebraic combinatorics community, it is nestled between two families of polytopes that certainly are not: the permutahedra $\mathfrak{S}_{\bullet}$ and associahedra $\mathcal{Y}_{\bullet}$. The maps $\mathfrak{S}_{\bullet} \twoheadrightarrow \mathcal{M}_{\bullet} \twoheadrightarrow \mathcal{Y}_{\bullet}$ reveal several new Hopf structures on tree-like objects nestled between the Hopf algebras $\mathfrak{S}Sym$ and $\mathcal{Y}Sym$. We begin their study here, showing that $\mathcal{M}Sym$ is a module over $\mathfrak{S}Sym$ and a Hopf module over $\mathcal{Y}Sym$. An elegant description of the coinvariants for $\mathcal{M}Sym$ over $\mathcal{Y}Sym$ is uncovered via a change of basis-using Möbius inversion in posets built on the $1$-skeleta of $\mathcal{M}_{\bullet}$. Our analysis uses the notion of an $\textit{interval retract}$ that should be of independent interest in poset combinatorics. It also reveals new families of polytopes, and even a new factorization of a known projection from the associahedra to hypercubes. Les multiplièdres $\mathcal{M}_{\bullet} = (\mathcal{M}_n)_{n \geq 1}$ forment une famille de polytopes en provenant de l'étude des catégories supérieures et de la théorie de l'homotopie. Tandis que les multiplihèdres sont peu connus dans la communauté de la combinatoire algébrique, ils sont nichés entre deux familles des polytopes qui sont bien connus: les permutahèdres $\mathfrak{S}_{\bullet}$ et les associahèdres $\mathcal{Y}_{\bullet}$. Les morphismes $\mathfrak{S}_{\bullet} \twoheadrightarrow \mathcal{M}_{\bullet} \twoheadrightarrow \mathcal{Y}_{\bullet}$ dévoilent plusieurs nouvelles structures de Hopf sur les arbres binaires entre les algèbres de Hopf $\mathfrak{S}Sym$ et $\mathcal{Y}Sym$. Nous commençons son étude ici, en démontrant que $\mathcal{M}Sym$ est un module sur $\mathfrak{S}Sym$ et un module de Hopf sur $\mathcal{Y}Sym$. Une description élégante des coinvariants de $\mathcal{M}Sym$ sur $\mathcal{Y}Sym$ est trouvée par moyen d'une change de base―en utilisant une inversion de Möbius dans certains posets construits sur le $1$-squelette de $\mathcal{M}_{\bullet}$. Notre analyse utilise la notion d'$\textit{interval retract}$, qui devrait être intéressante par soi-même dans la théorie des ensembles partiellement ordonnés. Notre analyse donne lieu également à des nouvelles familles des polytopes, et même une nouvelle factorisation d'une projection connue des associahèdres aux hypercubes.

scholarly journals A Combinatorial Approach to Multiplicity-Free Richardson Subvarieties of the Grassmannian

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Michelle Snider
Keyword(s):  
Combinatorial Proof ◽  
Combinatorial Approach ◽  
Möbius Inversion ◽  
International Audience ◽  
K Theory ◽  
Multiplicity Free ◽  
Grothendieck Polynomials

International audience We consider Buch's rule for K-theory of the Grassmannian, in the Schur multiplicity-free cases classified by Stembridge. Using a result of Knutson, one sees that Buch's coefficients are related to Möbius inversion. We give a direct combinatorial proof of this by considering the product expansion for Grassmannian Grothendieck polynomials. We end with an extension to the multiplicity-free cases of Thomas and Yong. On examine la règle de Buch pour la K-théorie de la variété grassmannienne dans les cas sans multiplicité de Schur, qui ont étés classifiés par Stembridge. En utilisant un résultat de Knutson, on démontre que les coefficients de Buch sont liés à l'inversion de Möbius. On en fait une preuve directe et combinatoire qui passe par le developpement de produits de polynômes de Grothendieck. Pour conclure, on donne une application de cette théorie aux cas sans multiplicité de Thomas et Yong.

scholarly journals A lattice point counting generalisation of the Tutte polynomial

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Amanda Cameron ◽  
Alex Fink
Keyword(s):  
Lattice Point ◽  
Tutte Polynomial ◽  
Minkowski Sum ◽  
Point Counting ◽  
Combinatorial Interpretation ◽  
Point Counts ◽  
Standard Simplex ◽  
International Audience

International audience The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion- contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids. This polynomial is constructed using lattice point counts in the Minkowski sum of the base polytope of a polymatroid and scaled copies of the standard simplex. We also show that, in the matroid case, our polynomial has coefficients of alternating sign, with a combinatorial interpretation closely tied to the Dawson partition.

Editorial Note

1946 ◽  
Vol 11 (1) ◽  
pp. 2-2
Keyword(s):  
Editorial Note ◽  
Speech Sounds ◽  
Left Hand ◽  
Hand Column ◽  
Right Hand ◽  
Infant Speech ◽  
The Right

In the article “Infant Speech Sounds and Intelligence” by Orvis C. Irwin and Han Piao Chen, in the December 1945 issue of the Journal, the paragraph which begins at the bottom of the left hand column on page 295 should have been placed immediately below the first paragraph at the top of the right hand column on page 296. To the authors we express our sincere apologies.

Eosinophilic vasculitis: an inhabitual and resistant manifestation of a vasculitis

VASA ◽  
2010 ◽  
Vol 39 (4) ◽  
pp. 344-348 ◽  
Author(s):  
Jandus ◽  
Bianda ◽  
Alerci ◽  
Gallino ◽  
Marone
Keyword(s):  
Skin Biopsy ◽  
Intravenous Iron ◽  
Small Vessel Vasculitis ◽  
High Dose ◽  
Raynaud Phenomenon ◽  
The Third ◽  
Left Elbow ◽  
Right Hand ◽  
The Right ◽  
Trunk Skin

A 55-year-old woman was referred because of diffuse pruritic erythematous lesions and an ischemic process of the third finger of her right hand. She was known to have anaemia secondary to hypermenorrhea. She presented six months before admission with a cutaneous infiltration on the left cubital cavity after a paravenous leakage of intravenous iron substitution. She then reported a progressive pruritic erythematous swelling of her left arm and lower extremities and trunk. Skin biopsy of a lesion on the right leg revealed a fibrillar, small-vessel vasculitis containing many eosinophils.Two months later she reported Raynaud symptoms in both hands, with a persistent violaceous coloration of the skin and cold sensation of her third digit of the right hand. A round 1.5 cm well-delimited swelling on the medial site of the left elbow was noted. The third digit of her right hand was cold and of violet colour. Eosinophilia (19 % of total leucocytes) was present. Doppler-duplex arterial examination of the upper extremities showed an occlusion of the cubital artery down to the palmar arcade on the right arm. Selective angiography of the right subclavian and brachial arteries showed diffuse alteration of the blood flow in the cubital artery and hand, with fine collateral circulation in the carpal region. Neither secondary causes of hypereosinophilia nor a myeloproliferative process was found. Considering the skin biopsy results and having excluded other causes of eosinophilia, we assumed the diagnosis of an eosinophilic vasculitis. Treatment with tacrolimus and high dose steroids was started, the latter tapered within 12 months and then stopped, but a dramatic flare-up of the vasculitis with Raynaud phenomenon occurred. A new immunosupressive approach with steroids and methotrexate was then introduced. This case of aggressive eosinophilic vasculitis is difficult to classify into the usual forms of vasculitis and constitutes a therapeutic challenge given the resistance to current immunosuppressive regimens.

Visual context differentially mediates left- and right-hand reaching movements

2009 ◽  
Author(s):  
Jos J. Adam ◽  
Susan Hoonhorst ◽  
Rick Muskens ◽  
Jay Pratt ◽  
Martin H. Fischer
Keyword(s):  
Reaching Movements ◽  
Visual Context ◽  
Right Hand ◽  
Left And Right

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.

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