scholarly journals A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants

2011 ◽  
Vol 227 (5) ◽  
pp. 2092-2106 ◽  
Author(s):  
J. Haglund
Keyword(s):  
Hilbert Series ◽  
Quotient Ring ◽  
Polynomial Expression

scholarly journals A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants (condensed version)

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
J. Haglund
Keyword(s):  
Hilbert Series ◽  
Quotient Ring ◽  
Rational Functions ◽  
Constant Term ◽  
Diagonal Action ◽  
Integer Coefficients ◽  
Polynomial Expression ◽  
International Audience ◽  
Special Case ◽  
Invent Math

International audience A special case of Haiman's identity [Invent. Math. 149 (2002), pp. 371–407] for the character of the quotient ring of diagonal coinvariants under the diagonal action of the symmetric group yields a formula for the bigraded Hilbert series as a sum of rational functions in $q,t$. In this paper we show how a summation identity of Garsia and Zabrocki for Macdonald polynomial Pieri coefficients can be used to transform Haiman's formula for the Hilbert series into an explicit polynomial in $q,t$ with integer coefficients. We also provide an equivalent formula for the Hilbert series as the constant term in a multivariate Laurent series. Un cas spécial de l'identité de Haiman [Invent. Math. \textbf149 (2002), pp. 371–407] pour le caractère de l'anneau quotient des coinvariants diagonaux sous l'action du groupe symétrique fournit une formule pour la série de Hilbert bigraduée comme somme de fonctions rationnelles en q,t. Dans cet article nous montrons comment une identité de sommation de Garsia et Zabrocki pour les coefficients de Pieri des polynômes de Macdonald peut être utilisée pour transformer la formule de Haiman pour la série de Hilbert en un polynôme explicite en q,t à coefficients entiers. Nous présentons également une formule équivalente pour la série de Hilbert comme terme constant d'une série de Laurent multivariée.

Development and evaluation of trans-Amadi groundwater parameters: The integration of finite element techniques

2019 ◽  
Author(s):  
Chem Int
Keyword(s):  
Finite Element ◽  
Polynomial Regression ◽  
Nitrate Concentration ◽  
Strong Relationship ◽  
Total Iron ◽  
Coefficient Of Determination ◽  
Port Harcourt ◽  
Polynomial Expression ◽  
The Relationship ◽  
Best Fit

Mathematical model was developed and evaluated to monitor and predict the groundwater characteristics of Trans-amadi region in Port Harcourt City. In this research three major components were considered such as chloride, total iron and nitrate concentration as well as the polynomial expression on the behavious on the concentration of each component was determined in terms of the equation of the best fit as well as the square root of the curve. The relationship between nitrate and distance traveled by Nitrate concentration by the model is given as Pc = 0.003x2 - 0.451x + 14.91with coefficient of determination, R² = 0.947, Chloride given as Pc = 0.000x2 - 0.071x + 2.343, R² = 0.951while that of Total Iron is given as Pc = 2E-05x2 - 0.003x + 0.110, R² = 0.930. All these show a strong relationship as established by Polynomial Regression Model. The finite element techniques are found useful in monitoring, predicting and simulating groundwater characteristics of Trans-amadi as well as the prediction on the variation on the parameters of groundwater with variation in time.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

scholarly journals The Hilbert Series of a Linear Symplectic Circle Quotient

2014 ◽  
Vol 23 (1) ◽  
pp. 46-65 ◽  
Author(s):  
Hans-Christian Herbig ◽  
Christopher Seaton
Keyword(s):  
Hilbert Series

scholarly journals 2, 12, 117, 1959, 45171, 1170086, …: a Hilbert series for the QCD chiral Lagrangian

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Lukáš Gráf ◽  
Brian Henning ◽  
Xiaochuan Lu ◽  
Tom Melia ◽  
Hitoshi Murayama
Keyword(s):  
Hilbert Series ◽  
Charge Conjugation ◽  
External Fields ◽  
Dynkin Diagrams ◽  
Non Linear ◽  
Very High ◽  
Chiral Lagrangian

Abstract We apply Hilbert series techniques to the enumeration of operators in the mesonic QCD chiral Lagrangian. Existing Hilbert series technologies for non-linear realizations are extended to incorporate the external fields. The action of charge conjugation is addressed by folding the $$ \mathfrak{su}(n) $$ su n Dynkin diagrams, which we detail in an appendix that can be read separately as it has potential broader applications. New results include the enumeration of anomalous operators appearing in the chiral Lagrangian at order p8, as well as enumeration of CP-even, CP-odd, C-odd, and P-odd terms beginning from order p6. The method is extendable to very high orders, and we present results up to order p16.(The title sequence is the number of independent C-even and P-even operators in the mesonic QCD chiral Lagrangian with three light flavors of quarks, at chiral dimensions p2, p4, p6, …)

m-potent commutators involving skew derivations and multilinear polynomials

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammad Ashraf ◽  
Sajad Ahmad Pary ◽  
Mohd Arif Raza
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Prime Rings ◽  
Skew Derivation ◽  
Martindale Quotient Ring ◽  
Multilinear Polynomials ◽  
The Right ◽  
The Relationship

AbstractLet {\mathscr{R}} be a prime ring, {\mathscr{Q}_{r}} the right Martindale quotient ring of {\mathscr{R}} and {\mathscr{C}} the extended centroid of {\mathscr{R}}. In this paper, we discuss the relationship between the structure of prime rings and the behavior of skew derivations on multilinear polynomials. More precisely, we investigate the m-potent commutators of skew derivations involving multilinear polynomials, i.e.,\big{(}[\delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})]\big{)}^{m}=[% \delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})],where {1<m\in\mathbb{Z}^{+}}, {f(x_{1},x_{2},\ldots,x_{n})} is a non-central multilinear polynomial over {\mathscr{C}} and δ is a skew derivation of {\mathscr{R}}.

scholarly journals The Coulomb and Higgs branches of $$ \mathcal{N} $$ = 1 theories of Class $$ {\mathcal{S}}_k $$

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Thomas Bourton ◽  
Alessandro Pini ◽  
Elli Pomoni
Keyword(s):  
String Theory ◽  
Special Class ◽  
Hilbert Series ◽  
Superconformal Index

Abstract Even though for generic $$ \mathcal{N} $$ N = 1 theories it is not possible to separate distinct branches of supersymmetric vacua, in this paper we study a special class of $$ \mathcal{N} $$ N = 1 SCFTs, these of Class $$ {\mathcal{S}}_k $$ S k for which it is possible to define Coulomb and Higgs branches precisely as for the $$ \mathcal{N} $$ N = 2 theories of Class $$ \mathcal{S} $$ S from which they descend. We study the BPS operators that parameterise these branches of vacua using the different limits of the superconformal index as well as the Coulomb and Higgs branch Hilbert Series. Finally, with the tools we have developed, we provide a check that six dimensional (1, 1) Little String theory can be deconstructed from a toroidal quiver in Class $$ {\mathcal{S}}_k $$ S k .

scholarly journals A note on Noetherian orders in Artinian rings

1979 ◽  
Vol 20 (2) ◽  
pp. 125-128 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Noetherian Ring ◽  
Quotient Ring ◽  
Triangular Matrix ◽  
Idempotent Element ◽  
Noetherian Rings ◽  
Central Idempotent ◽  
Matrix Rings ◽  
Regular Elements ◽  
Zero Divisors ◽  
Left And Right

Throughout this note, rings are associative with identity element but are not necessarily commutative. Let R be a left and right Noetherian ring which has an Artinian (classical) quotient ring. It was shown by S. M. Ginn and P. B. Moss [2, Theorem 10] that there is a central idempotent element e of R such that eR is the largest Artinian ideal of R. We shall extend this result, using a different method of proof, to show that the idempotent e is also related to the socle of R/N (where N, throughout, denotes the largest nilpotent ideal of R) and to the intersection of all the principal right (or left) ideals of R generated by regular elements (i.e. by elements which are not zero-divisors). There are many examples of left and right Noetherian rings with Artinian quotient rings, e.g. commutative Noetherian rings in which all the associated primes of zero are minimal together with full or triangular matrix rings over such rings. It was shown by L. W. Small that if R is any left and right Noetherian ring then R has an Artinian quotient ring if and only if the regular elements of R are precisely the elements c of R such that c + N is a regular element of R/N (for further details and examples see [5] and [6]). By the largest Artinian ideal of R we mean the sum of all the Artinian right ideals of R, and it was shown by T. H. Lenagan in [3] that this coincides in any left and right Noetherian ring R with the sum of all the Artinian left ideals of R.

scholarly journals Lattice constants and thermal expansion of H2O and D2O Ice Ih between 10 and 265 K. Addendum

2012 ◽  
Vol 68 (1) ◽  
pp. 91-91 ◽  
Author(s):  
K. Röttger ◽  
A. Endriss ◽  
Jörg Ihringer ◽  
S. Doyle ◽  
W. F. Kuhs
Keyword(s):  
Thermal Expansion ◽  
Unit Cell ◽  
Acta Cryst ◽  
Lattice Constants ◽  
Temperature Range ◽  
Polynomial Fit ◽  
Polynomial Expression ◽  
Cell Data ◽  
Cell Volumes ◽  
Volume Cell

In a previous paper we reported the lattice constants and thermal expansion of normal and deuterated ice Ih [Röttger et al. (1994). Acta Cryst. B50, 644–648]. Synchrotron X-ray powder diffraction data were used to obtain the lattice constants and unit-cell volumes of H2O and D2O ice Ih in the temperature range 15–265 K. A polynomial expression was given for the unit-cell volumes. It turns out that the coefficients quoted have an insufficient number of digits to faithfully reproduce the volume cell data. Here we provide a table with more significant digits. Moreover, we also provide the coefficients of a polynomial fit to the previously published a and c lattice constants of normal and deuterated ice Ih for the same temperature range.

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