Brenti's Open Problem on the Real-Rootedness of $q$-Eulerian Polynomials of Type $D$

2017 ◽  
Vol 31 (2) ◽  
pp. 918-926 ◽  
Author(s):  
Arthur L. B. Yang ◽  
Philip B. Zhang
Keyword(s):  
Open Problem ◽  
The Real ◽  
Eulerian Polynomials ◽  
Type D

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 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 New Eulerian Numbers of Type $D$

10.37236/5514 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Anna Borowiec ◽  
Wojciech Młotkowski
Keyword(s):  
Recurrence Relation ◽  
Generating Function ◽  
Probability Distributions ◽  
Type A ◽  
Eulerian Numbers ◽  
Eulerian Polynomials ◽  
Type D

We introduce a new array of type $D$ Eulerian numbers, different from that studied by Brenti, Chow and Hyatt. We find in particular the recurrence relation, Worpitzky formula and the generating function. We also find the probability distributions whose moments are Eulerian polynomials of type $A$, $B$ and $D$.

A case study in authorship attribution: The Mondrigo1

2021 ◽  
pp. 1-8
Author(s):  
Gerardo Sierra ◽  
Tonatiuh Hernández-García ◽  
Helena Gómez-Adorno ◽  
Gemma Bel-Enguix
Keyword(s):  
Unsupervised Learning ◽  
Open Problem ◽  
Authorship Attribution ◽  
The Real ◽  
Supervised And Unsupervised Learning ◽  
Single Author ◽  
Svm Algorithm ◽  
Mexican Government

In this paper, we present authorship attribution methods applied to ¡El Mondrigo! (1968), a controversial text supposedly created by order of the Mexican Government to defame a student strike. Up to now, although the authorship of the book has been attributed to several journalists and writers, it could not be demonstrated and remains an open problem. The work aims at establishing which one of the most commonly attributed writers is the real author. To do that, we implement methods based on stylometric features using textual distance, supervised, and unsupervised learning. The distance-based methods implemented in this work are Kilgarriff and Delta of Burrows, an SVM algorithm is used as the supervised method, and the k-means algorithm as the unsupervised algorithm. The applied methods were consistent by pointing out a single author as the most likely one.

Density of Real Zeros of the Tutte Polynomial

2018 ◽  
Vol 27 (3) ◽  
pp. 398-410
Author(s):  
SEONGMIN OK ◽  
THOMAS J. PERRETT
Keyword(s):  
Open Problem ◽  
Dense Subset ◽  
Tutte Polynomial ◽  
The Real ◽  
Real Zeros ◽  
Density Result ◽  
Positive Volume ◽  
Tutte Polynomials

The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the real zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane. This is the first density result for the real zeros of the Tutte polynomial in a region of positive volume. Our result almost confirms a conjecture of Jackson and Sokal except for one region which is related to an open problem on flow polynomials.

scholarly journals Generalization of the Harmonic Weighted Mean Via Pythagorean Invariance Identity and Application

2020 ◽  
Vol 34 (1) ◽  
pp. 104-122
Author(s):  
Peter Kahlig ◽  
Janusz Matkowski
Keyword(s):  
Open Problem ◽  
Functional Equations ◽  
Harmonic Mean ◽  
Weighted Mean ◽  
Geometric Means ◽  
The Real ◽  
Weighted Arithmetic ◽  
Real Functions ◽  
Pythagorean Harmony

AbstractUnder some simple conditions on the real functions f and g defined on an interval I ⊂ (0, ∞), the two-place functions Af (x, y) = f (x) + y − f (y) and {G_g}\left({x,y} \right) = {{g\left(x \right)} \over {g\left(y \right)}}y generalize, respectively, A and G, the classical weighted arithmetic and geometric means. In this note, basing on the invariance identity G ∘ (H, A) = G (equivalent to the Pythagorean harmony proportion), a suitable weighted extension Hf,g of the classical harmonic mean H is introduced. An open problem concerning the symmetry of Hf,g is proposed. As an application a method of effective solving of some functional equations involving means is presented.

scholarly journals On the Real-Rootedness of the Descent Polynomials of $(n-2)$-Stack Sortable Permutations

10.37236/4613 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Philip B. Zhang
Keyword(s):  
General Result ◽  
The Real ◽  
Real Zeros ◽  
Eulerian Polynomials

Bóna conjectured that the descent polynomials on $(n-2)$-stack sortable permutations have only real zeros. Brändén proved this conjecture by establishing a more general result. In this paper, we give another proof of Brändén's result by using the theory of $s$-Eulerian polynomials recently developed by Savage and Visontai.

Safety Culture in Functional and Axiological Approach

2021 ◽  
Vol 35 (35) ◽  
pp. 166-178
Author(s):  
Krystyna Skurjat
Keyword(s):  
Safety Culture ◽  
Open Problem ◽  
Social Order ◽  
Social Effects ◽  
Human Species ◽  
The Real ◽  
Moral Principles ◽  
The Social

It still remains an open problem as to which directions and methods are necessary to choose in the situations of risk, xenophobia, and intolerance, so as to settle disputes and resolve conflicts; it should be done in such a way as to have the desired social effects and to properly implement moral principles, including responsibility for the real future of one’s own, of one’s community, and of the whole human species. The author justifies the position that the processes of transformation of the social order should take a more remarkable account of the requirements of safety culture and praxeology recommendations.

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