scholarly journals Delannoy numbers and Legendre polytopes

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Gábor Hetyei
Keyword(s):  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Geometric Interpretation ◽  
Boundary Complex ◽  
Nous Montrons ◽  
Il Y A ◽  
Delannoy Numbers ◽  
International Audience

International audience We construct an $n$-dimensional polytope whose boundary complex is compressed and whose face numbers for any pulling triangulation are the coefficients of the powers of $(x-1)/2$ in the $n$-th Legendre polynomial. We show that the non-central Delannoy numbers count all faces in the lexicographic pulling triangulation that contain a point in a given open quadrant. We thus provide a geometric interpretation of a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago. The polytopes we construct are closely related to the root polytopes introduced by Gelfand, Graev, and Postnikov. \par No construisons un polytope de dimension $n$ dont le complexe de bord est comprimé et dont les nombres de faces dans toute triangulation "en tirant des sommets'' sont les coefficients des puissances de $(x-1)/2$ dans le $n$-ième polynôme de Legendre. Nous montrons que les nombres centraux de Delannoy comptent toutes les faces dans la triangulation "en tirant des sommets'' en ordre lexicographique qui contiennent un point dans un certain quadrant ouvert. Ainsi nous produisons une interprétation géométrique d'une relation entre les nombres de Delannoy centraux et les polynômes de Legendre, notée il y a 50 ans. Nos polytopes sont reliés intimement aux polytopes de racines introduits par Gelfand, Graev, et Postnikov.

scholarly journals Quivers and the Euclidean algebra (Extended abstract)

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Alistair Savage
Keyword(s):  
Moduli Spaces ◽  
Euclidean Group ◽  
Quiver Varieties ◽  
Nous Permet ◽  
Preprojective Algebras ◽  
Nous Montrons ◽  
Wild Representation Type ◽  
Il Y A ◽  
International Audience ◽  
Euclidean Algebra

International audience We show that the category of representations of the Euclidean group $E(2)$ is equivalent to the category of representations of the preprojective algebra of the quiver of type $A_{\infty}$. Furthermore, we consider the moduli space of $E(2)$-modules along with a set of generators. We show that these moduli spaces are quiver varieties of the type considered by Nakajima. These identifications allow us to draw on known results about preprojective algebras and quiver varieties to prove various statements about representations of $E(2)$. In particular, we show that $E(2)$ has wild representation type but that if we impose certain combinatorial restrictions on the weight decompositions of a representation, we obtain only a finite number of indecomposable representations. Nous montrons que la catégorie des représentations du groupe d'Euclide $E(2)$ est équivalente à la catégorie des représentations de l'algèbre préprojective de type $A_{\infty}$. De plus, nous considérons l'espace classifiant de modules de $E(2)$ avec un ensemble de générateurs. Nous montrons que ces espaces sont de variétés de carquois de Nakajima. Cette identification nous permet d'utiliser des résultats des algèbres préprojectives et des variétés de carquois pour prouver des affirmations sur des représentations de $E(2)$. En particulier, nous montrons que le type de représentations de $E(2)$ est sauvage mais si nous imposons des restrictions aux poids d'une représentation, il y a seulement un nombre fini de représentations qui ne sont pas décomposables.

scholarly journals On bijections between monotone rooted trees and the comb basis

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Fu Liu
Keyword(s):  
Lie Algebra ◽  
Rooted Trees ◽  
Close Connection ◽  
Free Lie Algebra ◽  
Nous Montrons ◽  
Il Y A ◽  
International Audience ◽  
Lie Brackets ◽  
Element Set

International audience Let $A$ be an $n$-element set. Let $\mathscr{L} ie_2(A)$ be the multilinear part of the free Lie algebra on $A$ with a pair of compatible Lie brackets, and $\mathscr{L} ie_2(A, i)$ the subspace of $\mathscr{L} ie_2(A)$ generated by all the monomials in $\mathscr{L} ie_2(A)$ with $i$ brackets of one type. The author and Dotsenko-Khoroshkin show that the dimension of $\mathscr{L} ie_2(A, i)$ is the size of $R_{A,i}$, the set of rooted trees on $A$ with $i$ decreasing edges. There are three families of bases known for $\mathscr{L} ie_2(A, i)$ the comb basis, the Lyndon basis, and the Liu-Lyndon basis. Recently, González D'León and Wachs, in their study of (co)homology of the poset of weighted partitions (which has close connection to $\mathscr{L} ie_2(A, i)$), asked whether there are nice bijections between $R_{A,i}$ and the comb basis or the Lyndon basis. We give a natural definition for " nice bijections " , and conjecture that there is a unique nice bijection between $R_{A,i}$ and the comb basis. We show the conjecture is true for the extreme cases where $i=0$, $n−1$. Soit $A$ un ensemble à $n$ éléments. Soit $\mathscr{L} ie_2(A)$ la partie multilinéaire de l'algèbre de Lie libre sur $A$ avec une paire de crochets de Lie compatibles et $\mathscr{L} ie_2(A, i)$ le sous-espace de$\mathscr{L} ie_2(A)$ généré par tous les monômes en $\mathscr{L} ie_2(A)$ avec $i$ supports d'un même type. L'auteur et Dotsenko-Khoroshkin montrent que la dimension de $\mathscr{L} ie_2(A, i)$ est la taille de la $R_{A,i}$, l'ensemble des arbres enracinés sur $A$ avec $i$ arêtes décroissantes. Il y a trois familles de bases connues pour $\mathscr{L} ie_2(A, i)$ : la base de peigne, la base Lyndon, et la base Liu-Lyndon. Récemment, Gonzalez, D' Léon et Wachs, dans leur étude de (co)-homologie de la poset des partitions pondérés, ont demandé si il y a des bijections jolies entre$R_{A,i}$, et la base de peigne ou la base Lyndon. Nous donnons une définition naturelle de "bijection jolie " , et un conjecture qu'il y a une seule bijection jolie entre $R_{A,i}$, et la base de peigne. Nous montrons que la conjecture est vraie pour les cas extrêmes: $i = 0$, et $n − 1$.

scholarly journals A Baxter class of a different kind, and other bijective results using tableau sequences ending with a row shape

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Sophie Burrill ◽  
Stephen Melczer ◽  
Marni Mishna
Keyword(s):  
Recent Result ◽  
Young Tableaux ◽  
Set Partitions ◽  
Nous Permet ◽  
Nous Montrons ◽  
Il Y A ◽  
International Audience

International audience Tableau sequences of bounded height have been central to the analysis of $k$-noncrossing set partitions and matchings. We show here that families of sequences that end with a row shape are particularly compelling and lead to some interesting connections. First, we prove that hesitating tableaux of height at most two ending with a row shape are counted by Baxter numbers. This permits us to define three new Baxter classes which, remarkably, do not obviously possess the antipodal symmetry of other known Baxter classes. Oscillating tableau of height bounded by $k$ ending in a row are in bijection with Young tableaux of bounded height 2$k$. We discuss this recent result, and somegenerating function implications. Many of our proofs are analytic in nature, so there are intriguing combinatorial bijections to be found. Les séquences de tableau de hauteur bornée sont au centre de l’analyse des partages et couplages. Nous montrons que les familles de séquences qui se terminent par une seule ligne sont particulièrement fascinantes. Tout d’abord, nous prouvons que les tableaux hésitants de hauteur au plus deux se terminant par une seule ligne sont dénombrés par les nombres de Baxter. Cela nous permet de définir trois nouvelles classes Baxter qui, remarquablement, ne possèdent évidemment pas la symétrie antipode des autres classes Baxter connus. Nous discutons le résultat récent qui dit que les tableaux oscillants de hauteur au plus $k$ se terminant dans une ligne sont en bijection avec les tableaux de Young de hauteur au plus 2$k$. Nos preuves sont analytiques, il y a donc des bijections combinatoiresintrigantes à trouver.

scholarly journals A diagrammatic approach to Kronecker squares

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Ernesto Vallejo
Keyword(s):  
Polynomial Function ◽  
Young Tableaux ◽  
Piecewise Polynomial ◽  
Piecewise Polynomial Function ◽  
Isomorphism Classes ◽  
Nous Montrons ◽  
Il Y A ◽  
International Audience ◽  
Kronecker Coefficients ◽  
Standard Young Tableaux

International audience In this paper we improve a method of Robinson and Taulbee for computing Kronecker coefficients and show that for any partition $\overline{ν}$ of $d$ there is a polynomial $k_{\overline{ν}}$ with rational coefficients in variables $x_C$, where $C$ runs over the set of isomorphism classes of connected skew diagrams of size at most $d$, such that for all partitions $\lambda$ of $n$, the Kronecker coefficient $\mathsf{g}(\lambda, \lambda, (n-d, \overline{ν}))$ is obtained from $k_{\overline{ν}}(x_C)$ substituting each $x_C$ by the number of $\lambda$-removable diagrams in $C$. We present two applications. First we show that for $\rho_{k} = (k, k-1,\ldots, 2, 1)$ and any partition $\overline{ν}$ of size $d$ there is a piecewise polynomial function $s_{\overline{ν}}$ such that $\mathsf{g}(\rho_k, \rho_k, (|\rho_k| - d, \overline{ν})) = s_{\overline{ν}} (k)$ for all $k$ and that there is an interval of the form $[c, \infty)$ in which $s_{\overline{ν}}$ is polynomial of degree $d$ with leading coefficient the number of standard Young tableaux of shape $\overline{ν}$. The second application is new stability property for Kronecker coefficients. Dans ce papier nous améliorons une méthode de Robinson-Taulbee pour calculer les coefficients de Kronecker et montrons que pour toute partition $\overline{ν}$ de $d$ il y a un polynôme $k_{\overline{ν}}$ avec coefficients rationnels dans les variables $x_C$, où $C$ est dans l’ensemble de classes d’isomorphisme des diagrammes gauches connexes de taille non plus que $d$, tel que pour toute partition $\lambda$ de $n$, le coefficient de Kronecker $\mathsf{g}(\lambda, \lambda, (n-d, \overline{ν}))$ est obtenu de $k_{\overline{ν}}(x_C)$ en substituant chaque $x_C$ pour le nombre de diagrammes $\lambda$-removables en $C$. Nous présentons deux applications. Premièrement nous montrons que pour $\rho_{k} = (k, k-1,\ldots, 2, 1)$ et une partition $\overline{ν}$ de taille $d$ il y a une fonction polynôme par morceaux $s_{\overline{ν}}$ tel que pour toute $k$ on a $\mathsf{g}(\rho_k, \rho_k, (|\rho_k| - d, \overline{ν})) = s_{\overline{ν}} (k)$ et qu'il y a une intervalle de la forme $[c, \infty)$ dans laquelle $s_{\overline{ν}}$ est polynôme de degré $d$ avec coefficient principal le nombre de tableaux de Young standard de forme $\overline{ν}$. La seconde application est une nouveau propriété de stabilité des coefficients de Kronecker.

scholarly journals Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function

2014 ◽  
Vol 2014 ◽  
pp. 1-24 ◽  
Author(s):  
David W. Pravica ◽  
Njinasoa Randriampiry ◽  
Michael J. Spurr
Keyword(s):  
Fourier Transform ◽  
Theta Function ◽  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Inverse Fourier Transform ◽  
Recursion Relations ◽  
Jacobi Theta Function ◽  
Convergence Results ◽  
The Family ◽  
Reciprocal Symmetry

The family ofnth orderq-Legendre polynomials are introduced. They are shown to be obtainable from the Jacobi theta function and to satisfy recursion relations and multiplicatively advanced differential equations (MADEs) that are analogues of the recursion relations and ODEs satisfied by thenth degree Legendre polynomials. Thenth orderq-Legendre polynomials are shown to have vanishingkth moments for0≤k<n, as does thenth degree truncated Legendre polynomial. Convergence results are obtained, approximations are given, a reciprocal symmetry is shown, and nearly orthonormal frames are constructed. Conditions are given under which a MADE remains a MADE under inverse Fourier transform. This is used to construct new wavelets as solutions of MADEs.

scholarly journals Macdonald polynomials at $t=q^k$

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Jean-Gabriel Luque
Keyword(s):  
Macdonald Polynomials ◽  
Nous Montrons ◽  
International Audience

International audience We investigate the homogeneous symmetric Macdonald polynomials $P_{\lambda} (\mathbb{X} ;q,t)$ for the specialization $t=q^k$. We show an identity relying the polynomials $P_{\lambda} (\mathbb{X} ;q,q^k)$ and $P_{\lambda} (\frac{1-q}{1-q^k}\mathbb{X} ;q,q^k)$. As a consequence, we describe an operator whose eigenvalues characterize the polynomials $P_{\lambda} (\mathbb{X} ;q,q^k)$. Nous nous intéressons aux propriétés des polynômes de Macdonald symétriques $P_{\lambda} (\mathbb{X} ;q,t)$ pour la spécialisation $t=q^k$. En particulier nous montrons une égalité reliant les polynômes $P_{\lambda} (\mathbb{X} ;q,q^k)$ et $P_{\lambda} (\frac{1-q}{1-q^k}\mathbb{X} ;q,q^k)$. Nous en déduisons la description d'un opérateur dont les valeurs propres caractérisent les polynômes $P_{\lambda} (\mathbb{X} ;q,q^k)$.

LEGENDRE POLYNOMIALS AS THE NORMALIZATION OF PHOTON-SUBTRACTED SQUEEZED STATES

2009 ◽  
Vol 24 (20) ◽  
pp. 1597-1603 ◽  
Author(s):  
HONG-YI FAN ◽  
LI-YUN HU ◽  
XUE-XIANG XU
Keyword(s):  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Hermite Polynomial ◽  
Vacuum State ◽  
Squeezed States ◽  
Squeezed State ◽  
Normalization Factor ◽  
Squeezed Vacuum State ◽  
Squeezed Vacuum ◽  
Excitation State

By converting the photon-subtracted squeezed state (PSSS) to a squeezed Hermite-polynomial excitation state we find that the normalization factor of PSSS is an m-order Legendre polynomial of the squeezing parameter, where m is the number of subtracted photons. Some new relations about the Legendre polynomials are obtained by this analysis. We also show that the PSSS can also be treated as a Hermite-polynomial excitation on squeezed vacuum state.

scholarly journals Random regression models to estimate genetic parameters for milk production of Guzerat cows using orthogonal Legendre polynomials

2014 ◽  
Vol 49 (5) ◽  
pp. 372-383 ◽  
Author(s):  
Maria Gabriela Campolina Diniz Peixoto ◽  
Daniel Jordan de Abreu Santos ◽  
Rusbel Raul Aspilcueta Borquis ◽  
Frank Ângelo Tomita Bruneli ◽  
João Cláudio do Carmo Panetto ◽  
...  
Keyword(s):  
Milk Production ◽  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Regression Models ◽  
Environmental Effect ◽  
Genetic Parameters ◽  
Genetic Effect ◽  
Random Regression ◽  
Additive Genetic Effect ◽  
Fifth Order

The objective of this work was to compare random regression models for the estimation of genetic parameters for Guzerat milk production, using orthogonal Legendre polynomials. Records (20,524) of test-day milk yield (TDMY) from 2,816 first-lactation Guzerat cows were used. TDMY grouped into 10-monthly classes were analyzed for additive genetic effect and for environmental and residual permanent effects (random effects), whereas the contemporary group, calving age (linear and quadratic effects) and mean lactation curve were analized as fixed effects. Trajectories for the additive genetic and permanent environmental effects were modeled by means of a covariance function employing orthogonal Legendre polynomials ranging from the second to the fifth order. Residual variances were considered in one, four, six, or ten variance classes. The best model had six residual variance classes. The heritability estimates for the TDMY records varied from 0.19 to 0.32. The random regression model that used a second-order Legendre polynomial for the additive genetic effect, and a fifth-order polynomial for the permanent environmental effect is adequate for comparison by the main employed criteria. The model with a second-order Legendre polynomial for the additive genetic effect, and that with a fourth-order for the permanent environmental effect could also be employed in these analyses.

scholarly journals Bayesian B-spline mapping for dynamic quantitative traits

2012 ◽  
Vol 94 (2) ◽  
pp. 85-95 ◽  
Author(s):  
JUN XING ◽  
JIAHAN LI ◽  
RUNQING YANG ◽  
XIAOJING ZHOU ◽  
SHIZHONG XU
Keyword(s):  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Simulated Dataset ◽  
Model Parameters ◽  
B Spline ◽  
B Splines ◽  
Bayesian Mapping ◽  
Bayesian Shrinkage ◽  
Dynamic Traits ◽  
Over Time

SummaryOwing to their ability and flexibility to describe individual gene expression at different time points, random regression (RR) analyses have become a popular procedure for the genetic analysis of dynamic traits whose phenotypes are collected over time. Specifically, when modelling the dynamic patterns of gene expressions in the RR framework, B-splines have been proved successful as an alternative to orthogonal polynomials. In the so-called Bayesian B-spline quantitative trait locus (QTL) mapping, B-splines are used to characterize the patterns of QTL effects and individual-specific time-dependent environmental errors over time, and the Bayesian shrinkage estimation method is employed to estimate model parameters. Extensive simulations demonstrate that (1) in terms of statistical power, Bayesian B-spline mapping outperforms the interval mapping based on the maximum likelihood; (2) for the simulated dataset with complicated growth curve simulated by B-splines, Legendre polynomial-based Bayesian mapping is not capable of identifying the designed QTLs accurately, even when higher-order Legendre polynomials are considered and (3) for the simulated dataset using Legendre polynomials, the Bayesian B-spline mapping can find the same QTLs as those identified by Legendre polynomial analysis. All simulation results support the necessity and flexibility of B-spline in Bayesian mapping of dynamic traits. The proposed method is also applied to a real dataset, where QTLs controlling the growth trajectory of stem diameters in Populus are located.

Numerical Solution of Eighth-order Boundary Value Problems by Using Legendre Polynomials

2017 ◽  
Vol 15 (02) ◽  
pp. 1750083 ◽  
Author(s):  
Anna Napoli ◽  
Waleed M. Abd-Elhameed
Keyword(s):  
Numerical Solution ◽  
Boundary Value Problems ◽  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Boundary Value ◽  
Operational Matrix ◽  
Harmonic Numbers ◽  
Eighth Order ◽  
Numerical Examples ◽  
Derivatives Of

The main aim of this paper is to present and analyze a numerical algorithm for the solution of eighth-order boundary value problems. The proposed solutions are spectral and they depend on a new operational matrix of derivatives of certain shifted Legendre polynomial basis, along with the application of the collocation method. The nonzero elements of the operational matrix are expressed in terms of the well-known harmonic numbers. Numerical examples provide favorable comparisons with other existing methods and ascertain the efficiency and applicability of the proposed algorithm.

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