scholarly journals CODES AND SHIFTED CODES OF PARTITIONS

2011 ◽  
Vol 21 (08) ◽  
pp. 1447-1462 ◽  
Author(s):  
J. T. HIRD ◽  
NAIHUAN JING ◽  
ERNEST STITZINGER
Keyword(s):  
Classical Result ◽  
Schur Functions ◽  
Combinatorial Identity ◽  
Bernstein Operators ◽  
Pieri Rules ◽  
Straightforward Consequence

In a recent paper, Carrell and Goulden found a combinatorial identity of the Bernstein operators that they then used to prove Bernstein's theorem. We show that this identity is a straightforward consequence of the classical result. We also show how a similar approach using the codes of partitions can be generalized from Schur functions to also include Schur Q-functions and derive the combinatorial formulation for both cases. We then apply them by examining the Littlewood–Richardson and Pieri rules.

scholarly journals CODES AND SHIFTED CODES

2012 ◽  
Vol 22 (06) ◽  
pp. 1250054
Author(s):  
J. T. HIRD ◽  
NAIHUAN JING ◽  
ERNEST STITZINGER
Keyword(s):  
Algebraic Structure ◽  
Schur Functions ◽  
Natural Setting ◽  
Bernstein Operators ◽  
Combinatorial Model

The action of the Bernstein operators on Schur functions was given in terms of codes by Carrell and Goulden (2011) and extended to the analog in Schur Q-functions in our previous work. We define a new combinatorial model of extended codes and show that both of these results follow from a natural combinatorial relation induced on codes. The new algebraic structure provides a natural setting for Schur functions indexed by compositions.

scholarly journals Pieri rules for Schur functions in superspace

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Miles Eli Jones ◽  
Luc Lapointe
Keyword(s):  
Generating Functions ◽  
Schur Functions ◽  
Macdonald Polynomials ◽  
International Audience ◽  
Pieri Rules

International audience The Schur functions in superspace $s_\Lambda$ and $\overline{s}_\Lambda$ are the limits $q=t= 0$ and $q=t=\infty$ respectively of the Macdonald polynomials in superspace. We present the elementary properties of the bases $s_\Lambda$ and $\overline{s}_\Lambda$ (which happen to be essentially dual) such as Pieri rules, dualities, monomial expansions, tableaux generating functions, and Cauchy identities. Les fonctions de Schur dans le superespace $s_\Lambda$ et $\overline{s}_\Lambda$ sont les limites $q=t= 0$ et $q=t=\infty$ respectivement des polynômes de Macdonald dans le superespace. Nous présentons les propriétés élémentaires des bases $s_\Lambda$ et $\overline{s}_\Lambda$ (qui sont essentiellement duales l'une de l'autre) tels que les règles de Pieri, la dualité, le développement en fonctions monomiales, les fonctions génératrices de tableaux et les identités de Cauchy.

scholarly journals Skew Littlewood―Richardson rules from Hopf algebras

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Thomas Lam ◽  
Aaron Lauve ◽  
Frank Sottile
Keyword(s):  
Hopf Algebras ◽  
Symplectic Group ◽  
Schur Functions ◽  
Affine Grassmannian ◽  
International Audience ◽  
Pieri Rules ◽  
Skew Schur Functions ◽  
Nous Utilisons

International audience We use Hopf algebras to prove a version of the Littlewood―Richardson rule for skew Schur functions, which implies a conjecture of Assaf and McNamara. We also establish skew Littlewood―Richardson rules for Schur $P-$ and $Q-$functions and noncommutative ribbon Schur functions, as well as skew Pieri rules for k-Schur functions, dual k-Schur functions, and for the homology of the affine Grassmannian of the symplectic group. Nous utilisons des algèbres de Hopf pour prouver une version de la règle de Littlewood―Richardson pour les fonctions de Schur gauches, qui implique une conjecture d'Assaf et McNamara. Nous établissons également des règles de Littlewood―Richardson gauches pour les $P-$ et $Q-$fonctions de Schur et les fonctions de Schur rubbans non commutatives, ainsi que des règles de Pieri gauches pour les $k-$fonctions de Schur, les $k-$fonctions de Schur duales, et pour l'homologie de la Grassmannienne affine du groupe symplectique.

scholarly journals Bernstein operators and super-Schur functions: combinatorial aspects

2018 ◽  
Vol 109 (6) ◽  
pp. 1303-1342
Author(s):  
L. Alarie-Vézina ◽  
O. Blondeau-Fournier ◽  
L. Lapointe ◽  
P. Mathieu
Keyword(s):  
Schur Functions ◽  
Bernstein Operators

scholarly journals Pieri rules for Schur functions in superspace

2017 ◽  
Vol 148 ◽  
pp. 57-115 ◽  
Author(s):  
Miles Jones ◽  
Luc Lapointe
Keyword(s):  
Schur Functions ◽  
Pieri Rules

Optimal Assembly of Systems Using Schur-Functions and Majorization.

1986 ◽  
Author(s):  
Emad El-Neweihi ◽  
Frank Proschan ◽  
Jayaram Sethuraman
Keyword(s):  
Schur Functions ◽  
Optimal Assembly

scholarly journals Lupaş type Bernstein operators on triangles based on quantum analogue

2021 ◽  
Vol 60 (6) ◽  
pp. 5909-5919
Author(s):  
Asif Khan ◽  
M.S. Mansoori ◽  
Khalid Khan ◽  
M. Mursaleen
Keyword(s):  
Bernstein Operators ◽  
Quantum Analogue

Versal topological stratification and the bifurcation geometry of map-germs of the plane

1990 ◽  
Vol 107 (1) ◽  
pp. 127-147 ◽  
Author(s):  
J. H. Rieger
Keyword(s):  
Differential Geometry ◽  
Classical Result ◽  
Smooth Surface ◽  
Critical Values ◽  
Central Projections ◽  
Apparent Contour ◽  
Smooth Map

The set of critical values of a map of the plane (of corank 1) can be regarded as the apparent contour of a smooth surface. It is a classical result of Whitney[13] that generically the apparent contour is a smooth fold curve with isolated cusps and transverse fold crossings. More recent classifications of smooth map-germs of the plane of low codimension (occurring in generic 2- or 3-parameter families), e.g. in [1, 5, 7], were motivated by a question in differential geometry: given any smooth surface which is generically embedded in ℝ3, produce a list of all possible (orthogonal or central) projections of such a surface.

scholarly journals The Rate of Convergence of Lupasq-Analogue of the Bernstein Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Heping Wang ◽  
Yanbo Zhang
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Continuous Functions ◽  
Bernstein Operators ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous

We discuss the rate of convergence of the Lupasq-analogues of the Bernstein operatorsRn,q(f;x)which were given by Lupas in 1987. We obtain the estimates for the rate of convergence ofRn,q(f)by the modulus of continuity off, and show that the estimates are sharp in the sense of order for Lipschitz continuous functions.

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