scholarly journals A Combinatorial Reciprocity Theorem for Hyperplane Arrangements

2010 ◽  
Vol 53 (1) ◽  
pp. 3-10 ◽  
Author(s):  
Christos A. Athanasiadis
Keyword(s):  
Characteristic Polynomial ◽  
Nonnegative Integer ◽  
Reciprocity Theorem ◽  
Hyperplane Arrangements ◽  
Finite Collection ◽  
Linear Forms ◽  
Reciprocity Law ◽  
Combinatorial Reciprocity Theorem

AbstractGiven a nonnegative integer m and a finite collection of linear forms on ℚd, the arrangement of affine hyperplanes in ℚd defined by the equations α(x) = k for α ∈ and integers k ∈ [–m,m] is denoted by . It is proved that the coefficients of the characteristic polynomial of are quasi-polynomials in m and that they satisfy a simple combinatorial reciprocity law.

scholarly journals Enumeration of Golomb Rulers and Acyclic Orientations of Mixed Graphs

10.37236/2741 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Matthias Beck ◽  
Tristram Bogart ◽  
Tu Pham
Keyword(s):  
Small Neighborhood ◽  
Fixed Number ◽  
Reciprocity Theorem ◽  
Additive Number Theory ◽  
Chromatic Polynomials ◽  
Acyclic Orientations ◽  
Research Activities ◽  
Golomb Rulers ◽  
Combinatorial Reciprocity Theorem ◽  
Mixed Graphs

A Golomb ruler is a sequence of distinct integers (the markings of the ruler) whose pairwise differences are distinct. Golomb rulers, also known as Sidon sets and $B_2$ sets, can be traced back to additive number theory in the 1930s and have attracted recent research activities on existence problems, such as the search for optimal Golomb rulers (those of minimal length given a fixed number of markings). Our goal is to enumerate Golomb rulers in a systematic way: we study$$g_m(t) := \# \left\{ {\bf x} \in {\bf Z}^{m+1} : \, 0 = x_0 < x_1 < \dots < x_m = t , \text{ all } x_j - x_k \text{ distinct} \right\} ,$$the number of Golomb rulers with $m+1$ markings and length $t$.Our main result is that $g_m(t)$ is a quasipolynomial in $t$ which satisfies a combinatorial reciprocity theorem: $(-1)^{m-1} g_m(-t)$ equals the number of rulers ${\bf x}$ of length $t$ with $m+1$ markings, each counted with its Golomb multiplicity, which measures how many combinatorially different Golomb rulers are in a small neighborhood of ${\bf x}$. Our reciprocity theorem can be interpreted in terms of certain mixed graphs associated to Golomb rulers; in this language, it is reminiscent of Stanley's reciprocity theorem for chromatic polynomials. Thus in the second part of the paper we develop an analogue of Stanley's theorem to mixed graphs, which connects their chromatic polynomials to acyclic orientations.

scholarly journals Hierarchical Zonotopal Power Ideals

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Matthias Lenz
Keyword(s):  
Characteristic Polynomial ◽  
Hilbert Series ◽  
Finite Sequence ◽  
Vector Spaces ◽  
Geometric Structures ◽  
Linear Forms ◽  
International Audience ◽  
Powers Of Linear Forms ◽  
Matroid Structure

International audience Zonotopal algebra deals with ideals and vector spaces of polynomials that are related to several combinatorial and geometric structures defined by a finite sequence of vectors. Given such a sequence $X$, an integer $k \geq -1$ and an upper set in the lattice of flats of the matroid defined by $X$, we define and study the associated $\textit{hierarchical zonotopal power ideal}$. This ideal is generated by powers of linear forms. Its Hilbert series depends only on the matroid structure of $X$. It is related to various other matroid invariants, $\textit{e. g.}$ the shelling polynomial and the characteristic polynomial. This work unifies and generalizes results by Ardila-Postnikov on power ideals and by Holtz-Ron and Holtz-Ron-Xu on (hierarchical) zonotopal algebra. We also generalize a result on zonotopal Cox modules due to Sturmfels-Xu. La théorie de l'algèbre "zonotopique'' s'occupe d'idéaux et d'espaces vectoriels de polynômes qui ont un rapport avec plusieurs structures combinatoires et géométriques définies par des suites finies de vecteurs. Étant donné une telle suite $X$, un nombre entier $k \geq -1$ et un ensemble supérieur dans le treillis des plans du matroïde défini par $X$, nous définissons et étudions l'$\textit{idéal hiérarchique zonotopique}$, engendré par des puissances de formes linéaires. Sa série de Hilbert dépend seulement de la structure matroïdale de $X$. Il existe des relations avec d'autres invariants de matroïdes, tels que le polynôme d'épluchage et le polynôme caractéristique. Ce travail unifie et généralise des résultats d'Ardila-Postnikov sur les idéaux de puissances et de Holtz-Ron et Holtz-Ron-Xu sur l'algèbre zonotopique (hiérarchique). Nous généralisons aussi un résultat sur les modules de Cox zonotopiques, dû à Sturmfels-Xu.

scholarly journals Freeness and multirestriction of hyperplane arrangements

2012 ◽  
Vol 148 (3) ◽  
pp. 799-806 ◽  
Author(s):  
Mathias Schulze
Keyword(s):  
Characteristic Polynomial ◽  
Hyperplane Arrangement ◽  
Hyperplane Arrangements

AbstractGeneralizing a result of Yoshinaga in dimension three, we show that a central hyperplane arrangement in 4-space is free exactly if its restriction with multiplicities to a fixed hyperplane of the arrangement is free and its reduced characteristic polynomial equals the characteristic polynomial of this restriction. We show that the same statement holds true in any dimension when imposing certain tameness hypotheses.

scholarly journals METRIC DIOPHANTINE APPROXIMATION FOR SYSTEMS OF LINEAR FORMS VIA DYNAMICS

2010 ◽  
Vol 06 (05) ◽  
pp. 1139-1168 ◽  
Author(s):  
DMITRY KLEINBOCK ◽  
GREGORY MARGULIS ◽  
JUNBO WANG
Keyword(s):  
Diophantine Approximation ◽  
Finite Collection ◽  
Linear Forms ◽  
Metric Diophantine Approximation ◽  
Divergence Estimates ◽  
Set Up

The goal of this paper is to generalize the main results of [21] and subsequent papers on metric with dependent quantities to the set-up of systems of linear forms. In particular, we establish "joint strong extremality" of arbitrary finite collection of smooth non-degenerate submanifolds of ℝn. The proofs are based on generalized quantitative non-divergence estimates for translates of measures on the space of lattices.

scholarly journals A Short Proof of the Rook Reciprocity Theorem

10.37236/1234 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Timothy Chow
Keyword(s):  
Present Author ◽  
Short Proof ◽  
Reciprocity Theorem ◽  
Combinatorial Proof ◽  
Reciprocity Law ◽  
Rook Numbers ◽  
New Formula

Rook numbers of complementary boards are related by a reciprocity law. A complicated formula for this law has been known for about fifty years, but recently Gessel and the present author independently obtained a much more elegant formula, as a corollary of more general reciprocity theorems. Here, following a suggestion of Goldman, we provide a direct combinatorial proof of this new formula.

scholarly journals Federer Measures, Good and Nonplanar Functions of Metric Diophantine Approximation

2017 ◽  
Vol 6 (2) ◽  
pp. 117
Author(s):  
Faiza Akram ◽  
Dongsheng Liu
Keyword(s):  
Diophantine Approximation ◽  
Finite Collection ◽  
Linear Forms ◽  
Metric Diophantine Approximation ◽  
Set Up ◽  
Polynomial Flows

The goal of this paper is to generalize the main results of [1] and subsequent papers on metric Diophantine approximation with dependent quantities to the set-up of systems of linear forms. In particular, we establish “joint strong extremality” of arbitrary finite collection of smooth nondegenerate submani- folds of .The proof was based on quantitative nondivergence estimates for quasi-polynomial flows on the space of lattices.

Application of the reciprocity theorem for the evaluation of vibrations generated by moving loads

2020 ◽  
pp. 33-40
Author(s):  
Nguyen Tron Tam ◽  
◽  
E.N. Kurbatsky ◽  
Nguyen Anh Tuan ◽  
◽  
...  
Keyword(s):  
Reciprocity Theorem ◽  
Moving Loads

Impact of Training and Development Programmes on the Productivity of Employees in the Banks

2016 ◽  
Vol 5 (1) ◽  
Author(s):  
Jaspreet Kaur
Keyword(s):  
Regression Analysis ◽  
Banking Sector ◽  
Human Resources Management ◽  
Training And Development ◽  
Linear Forms ◽  
Knowledge And Skills ◽  
Resources Management ◽  
Level Of Satisfaction ◽  
Log Linear ◽  
The Impact

Manpower training and development is an important aspect of human resources management which must be embarked upon either proactively or reactively to meet any change brought about in the course of time. Training is a continuous and perennial activity. It provides employees with the knowledge and skills to perform more effectively. The study examines the opinions of trainees regarding the impact of training and development programmes on the productivity of employees in the selected banks. To evaluate the impact of training and development programmes on productivity of banking sector, multiple regression analysis was employed in both log as well as log-linear forms. Also the impact of three sets of training i.e. objectives, methods and basics on level of satisfaction of respondents with the training was also examined through employing the regression analysis in the similar manner.

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