scholarly journals Enumerating Partial Latin Rectangles

10.37236/9093 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Raúl M. Falcón ◽  
Rebecca J. Stones
Keyword(s):  
Algebraic Geometry ◽  
Computational Methods ◽  
Latin Squares ◽  
Symmetric Polynomial ◽  
Chromatic Polynomial ◽  
Numerical Results ◽  
Polynomial Method ◽  
Symmetric Polynomials ◽  
Exact Size ◽  
Geometry Method

This paper deals with different computational methods to enumerate the set $\mathrm{PLR}(r,s,n;m)$ of $r \times s$ partial Latin rectangles on $n$ symbols with $m$ non-empty cells. For fixed $r$, $s$, and $n$, we prove that the size of this set is given by a symmetric polynomial of degree $3m$, and we determine the leading terms (the monomials of degree $3m$ through $3m-9$) using inclusion-exclusion. For $m \leqslant 13$, exact formulas for these symmetric polynomials are determined using a chromatic polynomial method. Adapting Sade's method for enumerating Latin squares, we compute the exact size of $\mathrm{PLR}(r,s,n;m)$, for all $r \leqslant s \leqslant n \leqslant 7$, and all $r \leqslant s \leqslant 6$ when $n=8$. Using an algebraic geometry method together with Burnside's Lemma, we enumerate isomorphism, isotopism, and main classes when $r \leqslant s \leqslant n \leqslant 6$. Numerical results have been cross-checked where possible.

scholarly journals A categorification of the chromatic symmetric polynomial

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Radmila Sazdanović ◽  
Martha Yip
Keyword(s):  
Symmetric Function ◽  
Symmetric Polynomial ◽  
Chromatic Polynomial ◽  
Khovanov Homology ◽  
Symmetric Polynomials ◽  
Nous Obtenons ◽  
Combinatorial Properties ◽  
International Audience ◽  
Chromatic Symmetric Function ◽  
Frobenius Series

International audience The Stanley chromatic polynomial of a graph $G$ is a symmetric function generalization of the chromatic polynomial, and has interesting combinatorial properties. We apply the ideas of Khovanov homology to construct a homology $H$<sub>*</sub>($G$) of graded $S_n$-modules, whose graded Frobenius series $Frob_G(q,t)$ reduces to the chromatic symmetric function at $q=t=1$. We also obtain analogues of several familiar properties of the chromatic symmetric polynomials in terms of homology. Le polynôme chromatique symétrique d’un graphe $G$ est une généralisation par une fonction symétrique du polynôme chromatique, et possède des propriétés combinatoires intéressantes. Nous appliquons les techniques de l’homologie de Khovanov pour construire une homologie $H$<sub>*</sub>($G$) de modules gradués $S_n$, dont la série bigraduée de Frobeniusse $Frob_G(q,t)$ réduit au polynôme chromatique symétrique à $q=t=1$. Nous obtenons également des analogies pour plusieurs propriétés connues des polynômes chromatiques en termes d’homologie.

scholarly journals Symmetric functions on spaces $\ell_p(\mathbb{{R}}^n)$ and $\ell_p(\mathbb{{C}}^n)$

2020 ◽  
Vol 12 (1) ◽  
pp. 5-16
Author(s):  
T.V. Vasylyshyn
Keyword(s):  
Linear Combination ◽  
Banach Spaces ◽  
Symmetric Functions ◽  
Symmetric Polynomial ◽  
Complex Numbers ◽  
Symmetric Polynomials ◽  
Algebraic Basis

This work is devoted to study algebras of continuous symmetric, that is, invariant with respect to permutations of coordinates of its argument, polynomials and $*$-polynomials on Banach spaces $\ell_p(\mathbb{R}^n)$ and $\ell_p(\mathbb{C}^n)$ of $p$-power summable sequences of $n$-dimensional vectors of real and complex numbers resp., where $1\leq p < +\infty.$ We construct the subset of the algebra of all continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n)$ such that every continuous symmetric polynomial on the space $\ell_p(\mathbb{R}^n)$ can be uniquely represented as a linear combination of products of elements of this set. In other words, we construct an algebraic basis of the algebra of all continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n).$ Using this result, we construct an algebraic basis of the algebra of all continuous symmetric $*$-polynomials on the space $\ell_p(\mathbb{C}^n).$ Results of the paper can be used for investigations of algebras, generated by continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n),$ and algebras, generated by continuous symmetric $*$-polynomials on the space $\ell_p(\mathbb{C}^n).$

scholarly journals Symmetric $*$-polynomials on $\mathbb C^n$

2018 ◽  
Vol 10 (2) ◽  
pp. 395-401
Author(s):  
T.V. Vasylyshyn
Keyword(s):  
Vector Space ◽  
Symmetric Polynomial ◽  
Vector Spaces ◽  
Symmetric Polynomials ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Cartesian Power ◽  
Elementary Symmetric Polynomials

$*$-Polynomials are natural generalizations of usual polynomials between complex vector spaces. A $*$-polynomial is a function between complex vector spaces $X$ and $Y,$ which is a sum of so-called $(p,q)$-polynomials. In turn, for nonnegative integers $p$ and $q,$ a $(p,q)$-polynomial is a function between $X$ and $Y,$ which is the restriction to the diagonal of some mapping, acting from the Cartesian power $X^{p+q}$ to $Y,$ which is linear with respect to every of its first $p$ arguments, antilinear with respect to every of its last $q$ arguments and invariant with respect to permutations of its first $p$ arguments and last $q$ arguments separately. In this work we construct formulas for recovering of $(p,q)$-polynomial components of $*$-polynomials, acting between complex vector spaces $X$ and $Y,$ by the values of $*$-polynomials. We use these formulas for investigations of $*$-polynomials, acting from the $n$-dimensional complex vector space $\mathbb{C}^n$ to $\mathbb{C},$ which are symmetric, that is, invariant with respect to permutations of coordinates of its argument. We show that every symmetric $*$-polynomial, acting from $\mathbb{C}^n$ to $\mathbb{C},$ can be represented as an algebraic combination of some "elementary" symmetric $*$-polynomials. Results of the paper can be used for investigations of algebras, generated by symmetric $*$-polynomials, acting from $\mathbb{C}^n$ to $\mathbb{C}.$

scholarly journals Differential operators and Higher Specht polynomials

2021 ◽  
Vol 2090 (1) ◽  
pp. 012096
Author(s):  
Ibrahim Nonkané ◽  
Léonard Todjihounde
Keyword(s):  
Symmetric Group ◽  
Polynomial Ring ◽  
Weyl Algebra ◽  
Differential Operators ◽  
Symmetric Groups ◽  
Symmetric Polynomial ◽  
Symmetric Polynomials ◽  
Polynomial Representation ◽  
Invariant Differential ◽  
Moser System

Abstract In this note, we study the action of the rational quantum Calogero-Moser system on polynomials rings. This a continuation of our paper [Ibrahim Nonkan 2021 J. Phys.: Conf. Ser. 1730 012129] in which we deal with the polynomial representation of the ring of invariant differential operators. Using the higher Specht polynomials we give a detailed description of the actions of the Weyl algebra associated with the ring of the symmetric polynomial C[x 1,..., xn]Sn on the polynomial ring C[x 1,..., xn ]. In fact, we show that its irreducible submodules over the ring of differential operators invariant under the symmetric group are its submodules generated by higher Specht polynomials over the ring of the symmetric polynomials. We end up studying the polynomial representation of the ring of differential operators invariant under the actions of products of symmetric groups by giving the generators of its simple components, thus we give a differential structure to the higher Specht polynomials.

scholarly journals INTEGRAL EQUATIONS AND ACTUARIAL RISK MANAGEMENT: SOME MODELS AND NUMERICS

2003 ◽  
Vol 8 (2) ◽  
pp. 143-154 ◽  
Author(s):  
A. Makroglou
Keyword(s):  
Risk Management ◽  
Differential Equations ◽  
Integral Equations ◽  
Computational Methods ◽  
Finite Time ◽  
Numerical Results ◽  
Insurance Company ◽  
Type Approximation ◽  
Actuarial Risk ◽  
Ruin Problem

The problem of the estimation of the probability R(z, t) (here t is time, z is initial reserve) of the finite time non‐ruin problem for a risk business such as an insurance company is considered, with respect to presenting models that have been used in the literature in the form of integral / integro-differential equations, reviewing some analytical and computational methods used for their solution, presenting numerical results obtained with one method (a global Lagrange type approximation in the z—space).

scholarly journals Recognition and Analysis of Image Patterns Based on Latin Squares by Means of Computational Algebraic Geometry

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 666
Author(s):  
Raúl M. Falcón
Keyword(s):  
Algebraic Geometry ◽  
Computer Algebra System ◽  
Latin Squares ◽  
Combinatorial Structures ◽  
New Approach ◽  
Algebraic Set ◽  
Computational Algebraic Geometry ◽  
Isomorphism Classes ◽  
Partial Transpose ◽  
Partial Latin Squares

With the particular interest of being implemented in cryptography, the recognition and analysis of image patterns based on Latin squares has recently arisen as an efficient new approach for classifying partial Latin squares into isomorphism classes. This paper shows how the use of a Computer Algebra System (CAS) becomes necessary to delve into this aspect. Thus, the recognition and analysis of image patterns based on these combinatorial structures benefits from the use of computational algebraic geometry to determine whether two given partial Latin squares describe the same affine algebraic set. This paper delves into this topic by focusing on the use of a CAS to characterize when two partial Latin squares are either partial transpose or partial isotopic.

scholarly journals Comparison of Actuator Line Method and Full Rotor Geometry Simulations of the Wake Field of a Tidal Stream Turbine

Water ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 560 ◽  
Author(s):  
Xiang-feng Lin ◽  
Ji-sheng Zhang ◽  
Yu-quan Zhang ◽  
Jing Zhang ◽  
Sheng Liu
Keyword(s):  
Numerical Approach ◽  
Numerical Results ◽  
Cfd Simulations ◽  
Wake Field ◽  
Line Method ◽  
Flow Features ◽  
Tidal Stream ◽  
Computational Fluid Dynamics Cfd ◽  
Geometry Method ◽  
Tidal Stream Turbine

This study aims to investigate the wake characteristics of a horizontal axis tidal stream turbine supported by a monopile using a numerical approach. Computational fluid dynamics (CFD) simulations based on the open source software OpenFOAM have been performed to enhance understanding of a turbine’s wake. The numerical simulations adopt both the actuator line method and the full rotor geometry method. The numerical results are found to be consistent with experimental data, although some discrepancies are observed at a distance of one rotor diameter downstream. Comparison of numerical results from both methods is performed. The results show that both methods can obtain important flow features and provide similar simulation in the wake of the turbine model. The actuator line method is able to give a better prediction in stream-wise velocity distribution, although it underestimates the turbulence intensity, circumferential velocity and vorticity magnitude slightly, compared with the full rotor geometry method. It is also found that the wake of the monopile and the rotor interact strongly in the downstream field, especially in the region immediately behind the structure. A strong interaction occurs within approximately two rotor diameters downstream.

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