scholarly journals Eulerian polynomials of type $D$ have only real roots

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Carla D. Savage ◽  
Mirkó Visontai
Keyword(s):  
Type B ◽  
Eulerian Polynomial ◽  
Eulerian Polynomials ◽  
Real Roots ◽  
Type D ◽  
International Audience

International audience We give an intrinsic proof of a conjecture of Brenti that all the roots of the Eulerian polynomial of type $D$ are real and a proof of a conjecture of Dilks, Petersen, and Stembridge that all the roots of the affine Eulerian polynomial of type $B$ are real, as well. Nous prouvons, de façon intrinsèque, une conjecture de Brenti affirmant que toutes les racines du polynôme eulérien de type $D$ sont réelles. Nous prouvons également une conjecture de Dilks, Petersen, et Stembridge que toutes les racines du polynôme eulérien affine de type $B$ sont réelles.

scholarly journals Gamma Positivity of the Descent Based Eulerian Polynomial in Positive Elements of Classical Weyl Groups

10.37236/9037 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Hiranya Kishore Dey ◽  
Sivaramakrishnan Sivasubramanian
Keyword(s):  
Coxeter Groups ◽  
Weyl Groups ◽  
Alternating Group ◽  
Type B ◽  
Eulerian Polynomial ◽  
Eulerian Polynomials ◽  
Type D ◽  
Positive Elements

The Eulerian polynomial $A_n(t)$ enumerating descents in $\mathfrak{S}_n$ is known to be gamma positive for all $n$. When enumeration is done over the type B and type D Coxeter groups, the type B and type D Eulerian polynomials are also known to be gamma positive for all $n$. We consider $A_n^+(t)$ and $A_n^-(t)$, the polynomials which enumerate descents in the alternating group $\mathcal{A}_n$ and in $\mathfrak{S}_n - \mathcal{A}_n$ respectively.  We show the following results about $A_n^+(t)$ and $A_n^-(t)$: both polynomials are gamma positive iff $n \equiv 0,1$ (mod 4). When $n \equiv 2,3$ (mod 4), both polynomials are not palindromic. When $n \equiv 2$ (mod 4), we show that {\sl two} gamma positive summands add up to give $A_n^+(t)$ and $A_n^-(t)$. When $n \equiv 3$ (mod 4), we show that {\sl three} gamma positive summands add up to give both $A_n^+(t)$ and $A_n^-(t)$.  We show similar gamma positivity results about the descent based type B and type D Eulerian polynomials when enumeration is done over the positive elements in the respective Coxeter groups. We also show that the polynomials considered in this work are unimodal.

scholarly journals Les polynômes eul\IeC èriens stables de type B

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Mirkó Visontai ◽  
Nathan Williams
Keyword(s):  
Linear Operator ◽  
Type B ◽  
Multivariate Polynomial ◽  
Eulerian Polynomial ◽  
Eulerian Polynomials ◽  
Signed Permutations ◽  
International Audience ◽  
Multiple Variables ◽  
Real Stability

International audience We give a multivariate analog of the type B Eulerian polynomial introduced by Brenti. We prove that this multivariate polynomial is stable generalizing Brenti's result that every root of the type B Eulerian polynomial is real. Our proof combines a refinement of the descent statistic for signed permutations with the notion of real stability—a generalization of real-rootedness to polynomials in multiple variables. The key is that our refined multivariate Eulerian polynomials satisfy a recurrence given by a stability-preserving linear operator. Nous prèsentons un raffinement multivariè d'un polynôme eulèrien de type B dèfini par Brenti. En prouvant que ce polynôme est stable nous gènèralisons un rèsultat de Brenti selon laquel chaque racine du polynôme eulèrien de type B est rèelle. Notre preuve combine un raffinement de la statistique des descentes pour les permutations signèes avec la stabilitè—une gènèralisation de la propriètè d'avoir uniquement des racines rèelles aux polynômes en plusieurs variables. La connexion est que nos polynômes eulèriens raffinès satisfont une rècurrence donnèe par un opèrateur linèaire qui prèserve la stabilitè.

scholarly journals The Real-rootedness of Eulerian Polynomials via the Hermite–Biehler Theorem

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Arthur L.B. Yang ◽  
Philip B. Zhang
Keyword(s):  
Linear Operators ◽  
Type B ◽  
The Real ◽  
Eulerian Polynomials ◽  
Hurwitz Stability ◽  
Type D ◽  
International Audience

International audience Based on the Hermite–Biehler theorem, we simultaneously prove the real-rootedness of Eulerian polynomials of type $D$ and the real-rootedness of affine Eulerian polynomials of type $B$, which were first obtained by Savage and Visontai by using the theory of $s$-Eulerian polynomials. We also confirm Hyatt’s conjectures on the inter-lacing property of half Eulerian polynomials. Borcea and Brändén’s work on the characterization of linear operators preserving Hurwitz stability is critical to this approach. Basé sur le théorème de Hermite–Biehler, nous prouvons simultanément les polynômes eulériens de type $D$ et les polynômes eulériens affine de type $B$ ont seulement racines réelle, qui sont d’abord obtenue par Savage et Visontai en utilisant le théorie des polynômes $s$-eulériens. Nous confirmons aussi les conjectures de Hyatt sur la propriété entrelacement de polynômes mi-eulériens. Le travail de Borcea et Brändén sur la caractérisation des opérateurs linéaires préservant la stabilité Hurwitz est essentielle à cette approche.

scholarly journals Recurrences for Eulerian Polynomials of Type B and Type D

2016 ◽  
Vol 20 (4) ◽  
pp. 869-881 ◽  
Author(s):  
Matthew Hyatt
Keyword(s):  
Type B ◽  
Eulerian Polynomials ◽  
Type D

scholarly journals Crossings and nestings in set partitions of classical types

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Martin Rubey ◽  
Christian Stump
Keyword(s):  
Nous Avons ◽  
Type A ◽  
The Other ◽  
Type B ◽  
Set Partition ◽  
Set Partitions ◽  
Type D ◽  
Nous Montrons ◽  
International Audience ◽  
The One

International audience In this extended abstract, we investigate bijections on various classes of set partitions of classical types that preserve openers and closers. On the one hand we present bijections for types $B$ and $C$ that interchange crossings and nestings, which generalize a construction by Kasraoui and Zeng for type $A$. On the other hand we generalize a bijection to type $B$ and $C$ that interchanges the cardinality of a maximal crossing with the cardinality of a maximal nesting, as given by Chen, Deng, Du, Stanley and Yan for type $A$. For type $D$, we were only able to construct a bijection between non-crossing and non-nesting set partitions. For all classical types we show that the set of openers and the set of closers determine a non-crossing or non-nesting set partition essentially uniquely. Dans ce résumé, nous étudions des bijections entre diverses classes de partitions d'ensemble de types classiques qui préservent les "openers'' et les "closers''. D'une part, nous présentons des bijections pour les types $B$ et $C$ qui échangent croisées et emboôtées, qui généralisent une construction de Kasraoui et Zeng pour le type $A$. D'autre part, nous généralisons une bijection pour le type $B$ et $C$ qui échange la cardinalité d'un croisement maximal avec la cardinalité d'un emboîtement maximal comme il a été fait par Chen, Deng, Du, Stanley et Yan pour le type $A$. Pour le type $D$, nous avons seulement construit une bijection entre les partitions non croisées et non emboîtées. Pour tout les types classiques, nous montrons que l'ensemble des "openers'' et l'ensemble des "closers'' déterminent une partition non croisées ou non emboîtées essentiellement de façon unique.

scholarly journals On the Monotone Column Permanent conjecture

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
James Haglund ◽  
Mirkó Visontai
Keyword(s):  
Nous Obtenons ◽  
Multivariate Polynomial ◽  
Eulerian Polynomials ◽  
Real Roots ◽  
International Audience ◽  
Univariate Polynomial ◽  
The Stability ◽  
Special Case ◽  
Nous Utilisons ◽  
General Method

International audience We discuss some recent progress on the Monotone Column Permanent (MCP) conjecture. We use a general method for proving that a univariate polynomial has real roots only, namely by showing that a corresponding multivariate polynomial is stable. Recent connections between stability of polynomials and the strong Rayleigh property revealed by Brändén allows for a computationally feasible check of stability for multi-affine polynomials. Using this method we obtain a simpler proof for the $n=3$ case of the MCP conjecture, and a new proof for the $n=4$ case. We also show a multivariate version of the stability of Eulerian polynomials for $n \leq 5$ which arises as a special case of the multivariate MCP conjecture. Nous présentons des développements récents concernant la conjecture Monotone Column Permanent (MCP). Nous utilisons une méthode générale pour prouver qu’un polynôme univarié a uniquement des racines réelles, c’est-à-dire que nous prouvons qu’un polynôme correspondant a plusieurs variables est stable. Les nouveaux liens, établis par Brändén, entre la stabilité des polynômes et la propriété forte de Rayleigh, permettent de vérifier facilement la stabilité de polynômes multi-affines. En utilisant cette méthode nous obtenons une preuve plus simple pour la conjecture MCP pour le cas $n=3$, et la première preuve pour le cas $n=4$. Nous présentons également une version multivariée de stabilité des polynômes d’Euler pour le cas $n \leq 5$, qui apparaît comme un cas spécial de la conjecture MCP multivariée.

scholarly journals Electrical network and Lie theory

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Yi Su
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Electrical Network ◽  
Type B ◽  
Electrical Networks ◽  
Combinatorial Properties ◽  
Type D ◽  
International Audience ◽  
Type C

International audience Curtis-Ingerman-Morrow studied the space of circular planar electrical networks, and classified all possible response matrices for such networks. Lam and Pylyavskyy found a Lie group $EL_{2n}$ whose positive part $(EL_{2n})_{\geq 0}$ naturally acts on the circular planar electrical network via some combinatorial description, where the action is inspired by the star-triangle transformation of the electrical networks. The Lie algebra $el_{2n}$ is semisimple and isomorphic to the symplectic algebra. In the end of their paper, they suggest a generalization of electrical Lie algebras to all finite Dynkin types. We give the structure of the type $B$ electrical Lie algebra $e_{b_{2n}}$. The nonnegative part $(E_{B_{2n}})_{\geq 0}$ of the corresponding Lie group conjecturally acts on a class of "mirror symmetric circular planar electrical networks". This class of networks has interesting combinatorial properties. Finally, we mention some partial results for type $C$ and $D$ electrical Lie algebras, where an analogous story needs to be developed. Curtis, Ingerman et Morrow ont étudié l’espace des réseaux électriques circulaires plans et ont classifié toutes les matrices de réponses possibles pour ces réseaux. Lam et Pylyavskyy ont trouvé un groupe de Lie $EL_{2n}$ dont la partie positive $(EL_{2n})_{\geq 0}$ agit naturellement sur le réseau électrique circulaire plan par une description combinatoire, où l’action est inspirée par la transformation étoile vers triangle des réseaux électriques. L’algèbre de Lie $el_{2n}$ est semi-simple et isomorphe à l’algèbre symplectique. A la fin de leur article, ils proposent une généralisation des algèbres de Lie électriques pour tous les types de Dynkin finis. Nous donnons la structure de l’algèbre de Lie électrique $e_{b_{2n}}$ du type $B$. La partie positive $(E_{B_{2n}})_{\geq 0}$ du groupe de Lie correspondant agit conjecturalement sur une famille de ”miroirs réseaux électriques circulaires symétriques plans”. Cette famille de réseaux a des propriétés combinatoires intéressantes. Nous donnons enfin quelques résultats partiels de l’algèbre de Lie électrique du type $C$ et du type $D$, où une étude analogue doit être développée.

scholarly journals CTNI-03. IMAGING FEATURES OF LEPTOMENINGEAL METASTASES OF NON-SMALL CELL LUNG CANCER: A SINGLE-CENTER REAL-WORLD STUDY

2020 ◽  
Vol 22 (Supplement_2) ◽  
pp. ii41-ii41
Author(s):  
Junjie Zhen ◽  
Lei Wen ◽  
Shaoqun Li ◽  
Mingyao Lai ◽  
Changguo Shan ◽  
...  
Keyword(s):  
Spinal Cord ◽  
Type A ◽  
Brain Parenchyma ◽  
Imaging Features ◽  
Type B ◽  
Type D ◽  
Mri Examination ◽  
Mri Features ◽  
Brain And Spinal Cord ◽  
The Brain

Abstract BACKGROUND According to EANO-ESMO clinical practice guidelines, the MRI findings of LM are divided into 4 types, namely linear enhancement (type A), nodular enhancement (type B), linear combined with nodular enhancement (type C), and sign of hydrocephalus (type D). METHODS The MRI features of brain and spinal cord in patients diagnosed with NSCLC-LM in Guangdong Sanjiu Brain Hospital from 2010 until 2019 were investigated, and then were classified into 4 types. The imaging features were analyzed. RESULTS A total of 80 patients were enrolled in the study. The median age of the patients was 53.5 years old, and the median time from the initial diagnosis to the confirmed diagnosis of LM was 11.6 months. The results of enhanced MRI examination of the brain in 79 cases showed that the number of cases with enhancements of type A, B, C and D were 50 (63.3%), 0, 26 (32.9%) and 3 (3.8%), respectively, and that LM with metastases to the brain parenchyma was found in 42 cases (53.2%). The results of enhanced MRI examination of spinal cord in 59 cases showed that there were only enhancements of type A and C in 40 cases (67.8%) and 3 cases (5.0%), and no enhancement sign in the other 16 cases (27.2%). CONCLUSION MRI examination of brain and spinal cord will improve the detection rate of LM. The MRI features of NSCLC-LM in real world are mainly characterized by the linear enhancements of brain and spinal cord, followed by linear combined with nodular enhancement. The enhancements of type B and type D are rare in clinic. Almost half of the patients have LM and metastases to the brain parenchyma. Therefore, the differentiation of tumor metastases is needed to be paid attention to for the early diagnosis and the formulation of reasonable treatment plans.

scholarly journals The effectiveness of TiO2 additions to mortar to maintain initial conditions in terms of its reflectance to solar radiation

2017 ◽  
Vol 17 (3) ◽  
pp. 39-56 ◽  
Author(s):  
Sérgio Roberto Andrade Dantas ◽  
Fulvio Vittorino ◽  
Kai Loh
Keyword(s):  
Solar Radiation ◽  
Ultraviolet Radiation ◽  
Initial Conditions ◽  
Type A ◽  
Just Noticeable Difference ◽  
Type B ◽  
Climate Conditions ◽  
Type D ◽  
Type C ◽  
Over Time

Abstract Contact of facades with degradation agents and direct incidence of ultraviolet radiation on external coatings make them more opaque over time, affecting their colour and reflectance characteristics. This study evaluated the effect of adding different TiO2 contents to mortars applied in concrete substrates in order to verify the reflectance maintenance on surfaces after exposure over time. Mortar with different concentrations of TiO2 (1%, 5%, 10%) were produced in relation to the total dry premix, added as a powder and compared to unpainted mortar without TiO2 (type "A") and painted mortar without TiO2 (type "B"), both used as a reference for colour and reflectance. Exposed over 16 months to climate conditions in São Paulo, regarding the maintenance of reflectance and solar radiation, the results showed that type "B" (0%TiO2) painted mortar presented the best performance. Type "C" (1%TiO2) and type "D" (5%TiO2) unpainted mortar remained more stable. Type "A" (0%TiO2) and type "E" (10%TiO2) unpainted mortar showed greater differences according to the Just Noticeable Difference (JND) range caused by dirt pick up.

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