scholarly journals Difference Families, Skew Hadamard Matrices, and Critical Groups of Doubly Regular Tournaments

10.37236/8753 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Venkata Raghu Tej Pantangi
Keyword(s):  
Hadamard Matrix ◽  
Abelian Groups ◽  
Difference Sets ◽  
Hadamard Matrices ◽  
Critical Group ◽  
The Other ◽  
Critical Groups ◽  
Difference Families ◽  
Skew Hadamard Difference Sets ◽  
Four Blocks

In this paper we investigate the structure of the critical groups of doubly-regular tournaments (DRTs) associated with skew Hadamard difference families (SDFs) with one, two, or four blocks. Brown and Ried found that the existence of a skew Hadamard matrix of order $n+1$ is equivalent to the existence of a DRT on $n$ vertices. A well known construction of a skew Hadamard matrix order $n$ is by constructing skew Hadamard difference sets in abelian groups of order $n-1$. The Paley skew Hadamard matrix is an example of one such construction. Szekeres and Whiteman constructed skew Hadamard matrices from skew Hadamard difference families with two blocks. Wallis and Whiteman constructed skew Hadamard matrices from skew Hadamard difference families with four blocks. In this paper we consider the critical groups of DRTs associated with skew Hadamard matrices constructed from skew Hadamard difference families with one, two or four blocks. We compute the critical groups of DRTs associated with skew Hadamard difference families with two or four blocks. We also compute the critical group of the Paley tournament and show that this tournament is inequivalent to the other DRTs we considered. Consequently we prove that the associated skew Hadamard matrices are not equivalent.   

scholarly journals A class of cyclic (v; k1, k2, k3; λ) difference families with v ≡ 3 (mod 4) a prime

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Dragomir Ž. Ðokovic ◽  
Ilias S. Kotsireas
Keyword(s):  
Difference Sets ◽  
Hadamard Matrices ◽  
Difference Families ◽  
First Time

AbstractWe construct several new cyclic (v; k1, k2, k3; λ) difference families, with v ≡ 3 (mod 4) a prime and λ = k1 + k2 + k3 − (3v − 1)/4. Such families can be used in conjunction with the well-known Paley-Todd difference sets to construct skew-Hadamard matrices of order 4v. Our main result is that we have constructed for the first time the examples of skew Hadamard matrices of orders 4 · 239 = 956 and 4 · 331 = 1324.

scholarly journals Some classes of Hadamard matrices with constant diagonal

1972 ◽  
Vol 7 (2) ◽  
pp. 233-249 ◽  
Author(s):  
Jennifer Wallis ◽  
Albert Leon Whiteman
Keyword(s):  
Prime Power ◽  
Abelian Groups ◽  
Difference Sets ◽  
Hadamard Matrices ◽  
Incidence Matrices

The concepts of circulant and backcirculant matrices are generalized to obtain incidence matrices of subsets of finite additive abelian groups. These results are then used to show the existence of skew-Hadamard matrices of order 8(4f+1) when f is odd and 8f + 1 is a prime power. This shows the existence of skew-Hadamard matrices of orders 296, 592, 1184, 1640, 2280, 2368 which were previously unknown.A construction is given for regular symmetric Hadamard matrices with constant diagonal of order 4(2m + 1)2 when a symmetric conference matrix of order 4m + 2 exists and there are Szekeres difference sets, X and Y, of size m satisfying x є X ⇒ −xє X, y є Y ⇒ −y єY.

scholarly journals Inequivalence of Skew Hadamard Difference Sets and Triple Intersection Numbers Modulo a Prime

10.37236/3762 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Koji Momihara
Keyword(s):  
Abelian Groups ◽  
Difference Sets ◽  
Intersection Numbers ◽  
Skew Hadamard Difference Sets ◽  
Triple Intersection ◽  
New Construction ◽  
Hadamard Difference Sets

Recently, Feng and Xiang found a new construction of skew Hadamard difference sets in elementary abelian groups. In this paper, we introduce a new invariant for equivalence of skew Hadamard difference sets, namely triple intersection numbers modulo a prime, and discuss inequivalence between Feng-Xiang skew Hadamard difference sets and the Paley difference sets. As a consequence, we show that their construction produces infinitely many skew Hadamard difference sets inequivalent to the Paley difference sets.

scholarly journals A Recursive Construction for Skew Hadamard Difference Sets

10.37236/9058 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Koji Momihara
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Difference Sets ◽  
Difference Set ◽  
Recursive Construction ◽  
Skew Hadamard Difference Sets ◽  
New Construction ◽  
Hadamard Difference Sets ◽  
Skew Hadamard Difference Set ◽  
Hadamard Difference Set

A major conjecture on the existence of abelian skew Hadamard difference sets is: if an abelian group $G$ contains a skew Hadamard difference set, then $G$ must be elementary abelian. This conjecture remains open in general. In this paper, we give a recursive construction for skew Hadamard difference sets in abelian (not necessarily elementary abelian) groups. The new construction can be considered as a result on the aforementioned conjecture: if there exists a counterexample to the conjecture, then there exist infinitely many counterexamples to it.

scholarly journals Constructions of Complex Hadamard Matrices via Tiling Abelian Groups

2007 ◽  
Vol 14 (03) ◽  
pp. 247-263 ◽  
Author(s):  
Máté Matolcsi ◽  
Júlia Réffy ◽  
Ferenc Szöllősi
Keyword(s):  
Information Theory ◽  
Necessary Conditions ◽  
Hadamard Matrix ◽  
Abelian Groups ◽  
Quantum Tomography ◽  
Hadamard Matrices ◽  
Quantum Information Theory ◽  
Current Interest ◽  
Parametric Families ◽  
Complex Hadamard Matrix

Applications in quantum information theory and quantum tomography have raised current interest in complex Hadamard matrices. In this note we investigate the connection between tiling of Abelian groups and constructions of complex Hadamard matrices. First, we recover a recent, very general construction of complex Hadamard matrices due to Dita [2] via a natural tiling construction. Then we find some necessary conditions for any given complex Hadamard matrix to be equivalent to a Dita-type matrix. Finally, using another tiling construction, due to Szabó [8], we arrive at new parametric families of complex Hadamard matrices of order 8, 12 and 16, and we use our necessary conditions to prove that these families do not arise with Dita's construction. These new families complement the recent catalogue [10] of complex Hadamard matrices of small order.

Relations Among Generalized Hadamard Matrices, Relative Difference Sets, and Maximal Length Linear Recurring Sequences

1963 ◽  
Vol 15 ◽  
pp. 42-48 ◽  
Author(s):  
A. T. Butson
Keyword(s):  
Incidence Matrix ◽  
Hadamard Matrix ◽  
Block Design ◽  
Difference Sets ◽  
Hadamard Matrices ◽  
Difference Set ◽  
Relative Difference ◽  
Block Designs ◽  
Linear Recurring Sequences ◽  
Balanced Incomplete Block

It was established in (5) that the existence of a Hadamard matrix of order 4t is equivalent to the existence of a symmetrical balanced incomplete block design with parameters v = 4t — 1, k = 2t — 1, and λ = t — 1. A block design is completely characterized by its so-called incidence matrix. The existence of a block design with parameters v, k, and λ such that the corresponding incidence matrix is cyclic was shown in (3) to be equivalent to the existence of a cyclic difference set with parameters v, k, and λ. For certain values of the parameters, Hadamard matrices, block designs, and difference sets do coexist.

scholarly journals Equivalence classes of inverse orthogonal and unit Hadamard matrices

1991 ◽  
Vol 44 (1) ◽  
pp. 109-115 ◽  
Author(s):  
R. Craigen
Keyword(s):  
Unit Circle ◽  
Prime Order ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Orthogonal Matrix ◽  
Equivalence Classes ◽  
The Other ◽  
Usual Sense ◽  
Orthogonal Matrices ◽  
Complex Valued

In 1867, Sylvester considered n × n matrices, (aij), with nonzero complex-valued entries, which satisfy (aij)(aij−1) = nI Such a matrix he called inverse orthogonal. If an inverse orthogonal matrix has all entries on the unit circle, it is a unit Hadamard matrix, and we have orthogonality in the usual sense. Any two inverse orthogonal (respectively, unit Hadamard) matrices are equivalent if one can be transformed into the other by a series of operations involving permutation of the rows and columns and multiplication of all the entries in any given row or column by a complex number (respectively a number on the unit circle). He stated without proof that there is exactly one equivalence class of inverse orthogonal matrices (and hence also of unit Hadamard matrices) in prime orders and that in general the number of equivalence classes is equal to the number of distinct factorisations of the order. In 1893 Hadamard showed this assertion to be false in the case of unit Hadamard matrices of non-prime order. We give the correct number of equivalence classes for each non-prime order, and orders ≤ 3, giving a complete, irredundant set of class representatives in each order ≤ 4 for both types of matrices.

On the SL(2, C)-representation rings of free abelian groups

2018 ◽  
Vol 167 (02) ◽  
pp. 229-247
Author(s):  
TAKAO SATOH
Keyword(s):  
Abelian Group ◽  
Upper Bound ◽  
Automorphism Groups ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Free Groups ◽  
The Other ◽  
Subsequent Research ◽  
Representation Rings ◽  
Free Abelian Groups

AbstractIn this paper, we study “the ring of component functions” of SL(2, C)-representations of free abelian groups. This is a subsequent research of our previous work [11] for free groups. We introduce some descending filtration of the ring, and determine the structure of its graded quotients.Then we give two applications. In [30], we constructed the generalized Johnson homomorphisms. We give an upper bound on their images with the graded quotients. The other application is to construct a certain crossed homomorphisms of the automorphism groups of free groups. We show that our crossed homomorphism induces Morita's 1-cocycle defined in [22]. In other words, we give another construction of Morita's 1-cocyle with the SL(2, C)-representations of the free abelian group.

scholarly journals What Affects Garment Lifespans? International Clothing Practices Based on a Wardrobe Survey in China, Germany, Japan, the UK, and the USA

2020 ◽  
Vol 12 (21) ◽  
pp. 9151
Author(s):  
Kirsi Laitala ◽  
Ingun Grimstad Klepp
Keyword(s):  
Quality Management ◽  
Environmental Impacts ◽  
Survey Data ◽  
Marketing Strategies ◽  
The Other ◽  
Multiple Regressions ◽  
The Usa ◽  
Logistic Regressions ◽  
Four Blocks ◽  
The Uk

Increasing the length of clothing lifespans is crucial for reducing the total environmental impacts. This article discusses which factors contribute to the length of garment lifespans by studying how long garments are used, how many times they are worn, and by how many users. The analysis is based on quantitative wardrobe survey data from China, Germany, Japan, the UK, and the USA. Variables were divided into four blocks related respectively to the garment, user, garment use, and clothing practices, and used in two hierarchical multiple regressions and two binary logistic regressions. The models explain between 11% and 43% of the variation in clothing lifespans. The garment use block was most indicative for the number of wears, while garment related properties contribute most to variation in the number of users. For lifespans measured in years, all four aspects were almost equally important. Some aspects that affect the lifespans of clothing cannot be easily changed (e.g., the consumer’s income, nationality, and age) but they can be used to identify where different measures can have the largest benefits. Several of the other conditions that affect lifespans can be changed (e.g., garment price and attitudes towards fashion) through quality management, marketing strategies, information, and improved consumer policies.

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