scholarly journals Moment graphs and KL-polynomials

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Martina Lanini
Keyword(s):  
New Techniques ◽  
International Audience ◽  
Combinatorial Data ◽  
Lusztig Polynomials

International audience Motivated by a result of Fiebig (2007), we categorify some properties of Kazhdan-Lusztig polynomials via sheaves on Bruhat moment graphs. In order to do this, we develop new techniques and apply them to the combinatorial data encoded in these moment graphs. Motivés par un résultat de Fiebig (2007), nous catégorisons certaines propriétés des polynômes de Kazhdan-Lusztig en utilisant faisceaux sur les graphes moment de Bruhat. Pour faire ça, nous développons de nouvelles techniques et les appliquons ensuite aux données combinatoires encodées dans ces graphes moment.

scholarly journals Peak algebras, paths in the Bruhat graph and Kazhdan-Lusztig polynomials

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Francesco Brenti ◽  
Fabrizio Caselli
Keyword(s):  
Combinatorial Formula ◽  
Quasisymmetric Functions ◽  
International Audience ◽  
Bruhat Graph ◽  
Lusztig Polynomials

International audience We obtain a nonrecursive combinatorial formula for the Kazhdan-Lusztig polynomials which holds in complete generality and which is simpler and more explicit than any existing one, and which cannot be linearly simplified. Our proof uses a new basis of the peak subalgebra of the algebra of quasisymmetric functions. On montre une formule combinatoire pour les polynômes de Kazhdan-Lusztig qui est valable en toute généralité. Cette formule est plus simple et plus explicite que toutes les autres formules connues; de plus, elle ne peut pas être simplifiée linéairement. La preuve utilise une nouvelle base pour la sous-algèbre des sommets de l’algèbre des fonctions quasi-symmetriques.

scholarly journals An immanant formulation of the dual canonical basis of the quantum polynomial ring

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Mark Skandera ◽  
Justin Lambright
Keyword(s):  
Polynomial Ring ◽  
Canonical Basis ◽  
Polynomial Rings ◽  
Dual Canonical Basis ◽  
International Audience ◽  
Quantum Polynomial ◽  
Lusztig Polynomials

International audience We show that dual canonical basis elements of the quantum polynomial ring in $n^2$ variables can be expressed as specializations of dual canonical basis elements of $0$-weight spaces of other quantum polynomial rings. Our results rely upon the natural appearance in the quantum polynomial ring of Kazhdan-Lusztig polynomials, $R$-polynomials, and certain single and double parabolic generalizations of these. Nous démontrons que des éléments de la base canonique duale de l'anneau quantique des polynômes en $n^2$ variables peuvent s'exprimer en termes des spécialisations d'éléments de la base canonique duale des espaces de poids $0$ d'autres anneaux quantiques. Nos résultats dépendent fortement de l'apparition naturelle des polynômes de Kazhdan-Lusztig, des $R$-polynômes, et de certaines généralisations simplement et doublement paraboliques de ces polynômes dans l'anneau quantique.

scholarly journals Permutations with Kazhdan-Lusztig polynomial $ P_id,w(q)=1+q^h$

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Alexander Woo
Keyword(s):  
International Audience ◽  
Lusztig Polynomials

International audience Using resolutions of singularities introduced by Cortez and a method for calculating Kazhdan-Lusztig polynomials due to Polo, we prove the conjecture of Billey and Braden characterizing permutations w with Kazhdan-Lusztig polynomial$ P_id,w(q)=1+q^h$ for some $h$. On démontre la conjecture de Billey et Braden sur les permutations w pour lesquelles le polynôme de Kazhdan-Lusztig $P_id,w(q)=1+q^h$ pour un entier $h$. On emploie une résolution des singularités présentées par Cortez et une méthode de Polo pour calculer ces polynômes.

scholarly journals Combination of tactile devices for data analytics

2019 ◽  
Vol Volume 8, Issue 1, Special... ◽  
Author(s):  
Gary Perelman ◽  
Marcos Serrano ◽  
Christophe Bortolaso ◽  
Célia Picard ◽  
Mustapha Derras ◽  
...  
Keyword(s):  
Data Analysis ◽  
Data Analytics ◽  
Interaction Techniques ◽  
Nous Proposons ◽  
New Techniques ◽  
Limited Size ◽  
Display Surface ◽  
International Audience ◽  
The Creation ◽  
Tactile Vocabulary

International audience Although ubiquitous data analysis is a promising approach, analyzing data in spreadsheets on tablets is a tedious task due to the limited size of the display and tactile vocabulary. In this article, we present the design and evaluation of new interaction techniques based on the combination of a tablet containing the data and a smartphone used as a mediator between the user and the tablet. To do this, we propose to use stacking gestures, i. e. to place a smartphone on top of a tablet. Stacking is an inexpensive, easy to implement, efficient and effective way to improve the analysis of data on tablets, increasing the vocabulary and broadening the display surface by using smartphones that are always available. We first explore stacking-based solutions to delimit the possible interaction vocabulary and present the manufacture of a conductive shell for smartphones. Then, we propose new techniques based on stacking to perform data analysis of a spreadsheet, i.e. the creation of pivot tables and their manipulation. We evaluate our stacking techniques against the tactile interactions provided by current mobile spreadsheet applications. Our studies reveal that some of our interaction techniques are 30% faster than touch to create pivot tables. Bien que l'analyse ubiquitaire de données soit une approche prometteuse, l'analyse des données dans des tableurs sur des tablettes est une tâche fastidieuse en raison de la taille limitée de l'affichage et du vocabulaire tactile. Dans cet article, nous présentons la conception et l'évaluation de nouvelles techniques d'interaction reposant sur la combinaison d'une tablette contenant les données et d'un smartphone utilisé comme médiateur entre l'utilisateur et la tablette. Pour ce faire, nous proposons d'utiliser des gestes de "stacking", c'est-à-dire de poser une arrête d'un smartphone sur l'écran de la tablette. Le stacking est un moyen peu coûteux, facile à mettre en oeuvre, efficace, et basé sur l'utilisation des smartphones toujours disponibles pour améliorer l'analyse des données sur des tablettes, en augmentant le vocabulaire utilisé et en élargissant la surface d'affichage. Nous explorons d'abord des solutions basées sur le stacking pour délimiter le vocabulaire d'interaction possible et présenter la fabrication d'une coque conductive pour smartphone. Ensuite, nous proposons de nouvelles techniques basées sur le stacking pour réaliser l'analyse de données d'un tableur, c'est-à-dire la création de tableaux croisés dynamiques et leur manipulation. Nous évaluons nos techniques de stacking par rapport aux interactions tactiles fournies par les applications de tableur mobiles actuelles. Nos études révèlent que certaines de nos techniques d'interaction sont 30% plus rapides que le toucher pour créer des tableaux croisés dynamiques.

scholarly journals Deodhar Elements in Kazhdan-Lusztig Theory

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Brant Jones
Keyword(s):  
Coxeter Groups ◽  
Pattern Avoidance ◽  
Weyl Groups ◽  
Schubert Varieties ◽  
Combinatorial Formula ◽  
Nous Obtenons ◽  
Integer Coefficients ◽  
Nous Montrons ◽  
International Audience ◽  
Lusztig Polynomials

International audience The Kazhdan-Lusztig polynomials for finite Weyl groups arise in representation theory as well as the geometry of Schubert varieties. It was proved very soon after their introduction that they have nonnegative integer coefficients, but no simple all positive interpretation for them is known in general. Deodhar has given a framework, which generally involves recursion, to express the Kazhdan-Lusztig polynomials in a very attractive form. We use a new kind of pattern-avoidance that can be defined for general Coxeter groups to characterize when Deodhar's algorithm yields a non-recursive combinatorial formula for Kazhdan-Lusztig polynomials $P_{x,w}(q)$ of finite Weyl groups. This generalizes results of Billey-Warrington which identified the $321$-hexagon-avoiding permutations, and Fan-Green which identified the fully-tight Coxeter groups. We also show that the leading coefficient known as $\mu (x,w)$ for these Kazhdan―Lusztig polynomials is always either $0$ or $1$. Finally, we generalize the simple combinatorial formula for the Kazhdan―Lusztig polynomials of the $321$-hexagon-avoiding permutations to the case when $w$ is hexagon avoiding and maximally clustered. Les polynômes de Kazhdan-Lusztig $P_{x,w}(q)$ des groupes de Weyl finis apparaissent en théorie des représentations, ainsi qu’en géométrie des variétés de Schubert. Il a été démontré peu après leur introduction qu’ils avaient des coefficients entiers positifs, mais on ne connaît toujours pas d’interprétation combinatoire simple de cette propriété dans le cas général. Deodhar a proposé un cadre donnant un algorithme, en général récursif, calculant des formules attractives pour les polynômes de Kazhdan-Lusztig. Billey-Warrington ont démontré que cet algorithme est non récursif lorsque$w$ évite les hexagones et les $321$ et qu’il donne des formules combinatoires simples. Nous introduisons une notion d’évitement de schémas dansles groupes de Coxeter quelconques nous permettant de généraliser les résultats de Billey-Warrington à tout groupe de Weyl fini. Nous montrons que le coefficient de tête $\mu (x,w)$ de ces polynômes de Kazhdan-Lusztig est toujours $0$ ou $1$. Cela généralise aussi des résultats de Fan-Greenqui identifient les groupes de Coxeter complètement serrés. Enfin, en type $A$, nous obtenons une classe plus large de permutations évitant la récursion.

scholarly journals Staircase diagrams and the enumeration of smooth Schubert varieties

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Edward Richmond ◽  
William Slofstra
Keyword(s):  
Data Structure ◽  
Finite Type ◽  
Schubert Varieties ◽  
International Audience ◽  
Combinatorial Data

International audience In this extended abstract, we give a complete description and enumeration of smooth and rationally smooth Schubert varieties in finite type. In particular, we show that rationally smooth Schubert varieties are in bijection with a new combinatorial data structure called staircase diagrams.

Stellar Evolution

1962 ◽  
Vol 11 (02) ◽  
pp. 137-143
Author(s):  
M. Schwarzschild
Keyword(s):  
Solar System ◽  
Stellar Evolution ◽  
Space Craft ◽  
New Techniques ◽  
Substantial Body ◽  
Individual Stars ◽  
The Past ◽  
Evolution Of Galaxies ◽  
History Of ◽  
Fast Flow

It is perhaps one of the most important characteristics of the past decade in astronomy that the evolution of some major classes of astronomical objects has become accessible to detailed research. The theory of the evolution of individual stars has developed into a substantial body of quantitative investigations. The evolution of galaxies, particularly of our own, has clearly become a subject for serious research. Even the history of the solar system, this close-by intriguing puzzle, may soon make the transition from being a subject of speculation to being a subject of detailed study in view of the fast flow of new data obtained with new techniques, including space-craft.

Microstructural characterization of CoPtCr/Cr thin film disks by cross-section TEM and elongated probe micro-diffraction

1991 ◽  
Vol 49 ◽  
pp. 762-763
Author(s):  
M.A. Parker ◽  
K.E. Johnson ◽  
C. Hwang ◽  
A. Bermea
Keyword(s):  
Magnetic Recording ◽  
Electron Probe ◽  
Microstructural Characterization ◽  
Crystallographic Structure ◽  
Recording Media ◽  
Layer By Layer ◽  
Cross Sectional ◽  
New Techniques ◽  
Polished Side

We have reported the dependence of the magnetic and recording properties of CoPtCr recording media on the thickness of the Cr underlayer. It was inferred from XRD data that grain-to-grain epitaxy of the Cr with the CoPtCr was responsible for the interaction observed between these layers. However, no cross-sectional TEM (XTEM) work was performed to confirm this inference. In this paper, we report the application of new techniques for preparing XTEM specimens from actual magnetic recording disks, and for layer-by-layer micro-diffraction with an electron probe elongated parallel to the surface of the deposited structure which elucidate the effect of the crystallographic structure of the Cr on that of the CoPtCr.XTEM specimens were prepared from magnetic recording disks by modifying a technique used to prepare semiconductor specimens. After 3mm disks were prepared per the standard XTEM procedure, these disks were then lapped using a tripod polishing device. A grid with a single 1mmx2mm hole was then glued with M-bond 610 to the polished side of the disk.

Computer averaging of lattice images of poly-4-methyl-pentene-1 (P4mp1): Noise created artifacts

1987 ◽  
Vol 45 ◽  
pp. 388-389
Author(s):  
P. Pradère ◽  
J.F. Revol ◽  
R. St. John Manley
Keyword(s):  
Low Dose ◽  
Polymer Concentration ◽  
Processing System ◽  
Limiting Factor ◽  
Dilute Solution ◽  
New Techniques ◽  
Dose Unit ◽  
Lattice Images ◽  
Dose Imaging ◽  
Imaging Conditions

Although radiation damage is the limiting factor in HREM of polymers, new techniques based on low dose imaging at low magnification have permitted lattice images to be obtained from very radiation sensitive polymers such as polyethylene (PE). This paper describes the computer averaging of P4MP1 lattice images. P4MP1 is even more sensitive than PE (total end point dose of 27 C m-2 as compared to 100 C m-2 for PE at 120 kV). It does, however, have the advantage of forming flat crystals from dilute solution and no change in d-spacings is observed during irradiation.Crystals of P4MP1 were grown at 60°C in xylene (polymer concentration 0.05%). Electron microscopy was performed with a Philips EM 400 T microscope equipped with a Low Dose Unit and operated at 120 kV. Imaging conditions were the same as already described elsewhere. Enlarged micrographs were digitized and processed with the Spider image processing system.

Serial thin sectioning for pre- and post-embedding immunohistochemistry: Achieving consistent and reliable results for a 3-D reconstruction

1993 ◽  
Vol 51 ◽  
pp. 328-329
Author(s):  
Antonia M. Milroy
Keyword(s):  
Nervous System ◽  
Computer Assisted ◽  
Serial Sectioning ◽  
Electron Microscopic ◽  
High Quality ◽  
New Techniques ◽  
3 Dimensional ◽  
Multiple User ◽  
Thin Sectioning ◽  
Subcellular Level

In recent years many new techniques and instruments for 3-Dimensional visualization of electron microscopic images have become available. Higher accelerating voltage through thicker sections, photographed at a tilt for stereo viewing, or the use of confocal microscopy, help to analyze biological material without the necessity of serial sectioning. However, when determining the presence of neurotransmitter receptors or biochemical substances present within the nervous system, the need for good serial sectioning (Fig. 1+2) remains. The advent of computer assisted reconstruction and the possibility of feeding information from the specimen viewing chamber directly into a computer via a camera mounted on the electron microscope column, facilitates serial analysis. Detailed information observed at the subcellular level is more precise and extensive and the complexities of interactions within the nervous system can be further elucidated.We emphasize that serial ultra thin sectioning can be performed routinely and consistently in multiple user electron microscopy laboratories. Initial tissue fixation and embedding must be of high quality.

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