scholarly journals Residual $q$-Fano Planes and Related Structures

10.37236/7107 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Tuvi Etzion ◽  
Niv Hooker
Keyword(s):  
Block Design ◽  
Infinite Family ◽  
Steiner System ◽  
Interesting Case ◽  
Common Belief ◽  
Steiner Systems ◽  
Fano Plane ◽  
Open Problems ◽  
One Step ◽  
Existence Question

One of the most intriguing problems for $q$-analogs of designs, is the existence question of an infinite family of $q$-Steiner systems that are not spreads. In particular the most interesting case is the existence question for the $q$-analog of the Fano plane, known also as the $q$-Fano plane. These questions are in the front line of open problems in block design. There was a common belief and a conjecture that such structures do not exist. Only recently, $q$-Steiner systems were found for one set of parameters. In this paper, a definition for the $q$-analog of the residual design is presented. This new definition is different from previous known definition, but its properties reflect better the $q$-analog properties. The existence of a design with the parameters of the residual $q$-Steiner system in general and the residual $q$-Fano plane in particular are examined. We construct different residual $q$-Fano planes for all $q$, where $q$ is a prime power. The constructed structure is just one step from a construction of a $q$-Fano plane.

scholarly journals A New Approach for Examining $q$-Steiner Systems

10.37236/7106 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Tuvi Etzion
Keyword(s):  
Necessary Conditions ◽  
Infinite Family ◽  
Steiner System ◽  
Uniform Structure ◽  
Steiner Systems ◽  
Fano Plane ◽  
New Approach ◽  
New Family ◽  
New Type ◽  
Existence Question

One of the most intriguing problems in $q$-analogs of designs and codes is the existence question of an infinite family of $q$-analog of Steiner systems (spreads not included) in general, and the existence question for the $q$-analog of the Fano plane in particular.We exhibit a completely new method to attack this problem. In the process we define a new family of designs whose existence is implied by the existence of $q$-Steiner systems, but could exist even if the related $q$-Steiner systems do not exist.The method is based on a possible system obtained by puncturing all the subspaces of the $q$-Steiner system several times. We define  the punctured system as a new type of design and enumerate the number of subspaces of various types that it might have. It will be evident that its existence does not imply the existence of the related $q$-Steiner system. On the other hand, this type of design demonstrates how close can we get to the related $q$-Steiner system.Necessary conditions for the existence of such designs are presented. These necessary conditions will be also necessary conditions for the existence of the related $q$-Steiner system. Trivial and nontrivial direct constructions and a nontrivial recursive construction for such designs are given. Some of the designs have a symmetric structure, which is uniform in the dimensions of the existing subspaces in the system. Most constructions are based on this uniform structure of the design or its punctured designs.

A Theorem on Steiner Systems

1970 ◽  
Vol 22 (5) ◽  
pp. 1010-1015 ◽  
Author(s):  
N. S. Mendelsohn
Keyword(s):  
Block Design ◽  
Steiner System ◽  
Incomplete Block ◽  
Steiner Systems ◽  
Balanced Incomplete Block Design ◽  
Tactical Configuration ◽  
Incomplete Block Design ◽  
Balanced Incomplete Block ◽  
The Given ◽  
Generalized Steiner System

1. Definitions and notation. A generalized Steiner system (t-design, tactical configuration) with parameters t, λt, k, v is a system (T, B), where T is a set of v elements, B is a set of blocks each of which is a k-subset of T (but note that blocks bi and bj may be the same k-subset of T) and such that every set of t elements of T belongs to exactly λt of the blocks. If we put λt = u we denote by Su(t, k, v) the collection of all systems with these parameters. Thus Q ∈ Su(t, k, v) means Q = (T, B) is a system with the given parameters. If λt = u = 1, we write S(t, k, v) instead of S1(t, k, v) and refer to the system as a Steiner system. If t = 2, the system is called a balanced incomplete block design.

scholarly journals Open problems in traceability: from raw materials to finished food products

2013 ◽  
Vol 44 (2s) ◽  
Author(s):  
Lorenzo Comba ◽  
Fabrizio Dabbene ◽  
Paolo Gay ◽  
Cristina Tortia
Keyword(s):  
Supply Chain ◽  
Raw Materials ◽  
Product Recalls ◽  
Open Problems ◽  
Food Traceability ◽  
Clear Sign ◽  
One Step ◽  
Risk Notification ◽  
Production Period ◽  
The Impact

Even though the main EU regulations concerning food traceability have already entered to force since many years, we still remark very wide and impacting product recalls, which often involve simultaneously large territories and many countries. This is a clear sign that current traceability procedures and systems, when implemented with the only aim of respecting mandatory policies, are not effective, and that there are some aspects that are at present underestimated, and therefore should be attentively reconsidered. In particular, the sole adoption of the so-called “one step back-one step forward traceability” to comply the EC Regulation 178/2002, where every actor in the chain handles merely the data coming from his supplier and those sent to his client, is in fact not sufficient to control and to limit the impact of a recall action after a risk notification. Recent studies on lots dispersion and routing demonstrate that each stakeholder has to plan his activities (production, transformation or distribution) according to specific criteria that allow pre-emptively estimating and limiting the range action of a possible recall. Moreover, these new and very recently proposed techniques still present some limits; first of all the problem of traceability of bulk products (e.g. liquids, powders, grains, crystals) during production phases that involve mixing operations of several lots of different/same materials. In fact, current traceability practices are in most cases unable to deal efficiently with this kind of products, and, in order to compensate the lack of knowledge about lot composition, typically resort to the adoption of very large lots, based for instance on a considered production period. Aim of this paper is to present recent advances in the design of supply chain traceability systems, discussing problems that are still open and are nowadays subject of research.

One step up and two steps back? The Italian debate on secularization, heteronormativity and LGBTQ citizenship

2018 ◽  
Vol 65 (5) ◽  
pp. 591-607 ◽  
Author(s):  
Elisa Bellè ◽  
Caterina Peroni ◽  
Elisa Rapetti
Keyword(s):  
Critical Discourse Analysis ◽  
Political Conflict ◽  
Interesting Case ◽  
Critical Discourse ◽  
Political Events ◽  
The Family ◽  
One Step ◽  
Definition Of ◽  
Shed Light

The aim of this article is to furnish insights of the Italian public debate on the recognition of LGBTQ rights, which can be understood as an interesting case study of the complex relationship between (multi)secularisation processes and re/definition of citizenship models. More specifically, the article analyses two political events related to this debate that took place in Rome in June 2015. The first is the Family Day demonstration, promoted by conservative Catholic groups; the second is the LGBTQ Pride parade, promoted by various gay, lesbian and transsexual/gender associations. We analyse the official statements issued by the two organising committees of the demonstrations, adopting the framework and methods of the Critical Discourse Analysis. Above and beyond an evident political conflict between the two discourses, we try to shed light on their mutual construction on the basis of what we call ‘naturalization’ and ‘universalization’ processes.

scholarly journals Analysis of Some Properties from Construction of Finite Projective Planes of Order 2 and 3

CAUCHY ◽  
2016 ◽  
Vol 4 (3) ◽  
pp. 131
Author(s):  
Vira Hari Krisnawati ◽  
Corina Karim
Keyword(s):  
Automorphism Group ◽  
Projective Plane ◽  
Symmetric Group ◽  
Block Design ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Steiner System ◽  
Incidence Relation ◽  
Finite Projective Planes ◽  
Set Of Points

<p class="abstract"><span lang="IN">In combinatorial mathematics, a Steiner system is a type of block design. Specifically, a Steiner system <em>S</em>(<em>t</em>, <em>k</em>, <em>v</em>) is a set of <em>v</em> points and <em>k</em> blocks which satisfy that every <em>t</em>-subset of <em>v</em>-set of points appear in the unique block. It is well-known that a finite projective plane is one examples of Steiner system with <em>t</em> = 2, which consists of a set of points and lines together with an incidence relation between them and order 2 is the smallest order.</span></p><p class="abstract"><span lang="IN">In this paper, we observe some properties from construction of finite projective planes of order 2 and 3. Also, we analyse the intersection between two projective planes by using some characteristics of the construction and orbit of projective planes over some representative cosets from automorphism group in the appropriate symmetric group.</span></p>

From the Outside In

2015 ◽  
Author(s):  
Derek Smith
Keyword(s):  
Positive Integer ◽  
Infinite Family ◽  
Open Problems ◽  
The Family

This chapter discusses Slothouber–Graatsma–Conway puzzle, which asks one to assemble six 1 × 2 × 2 pieces and three 1 × 1 × 1 pieces into the shape of a 3 × 3 × 3 cube. The puzzle has been generalized to larger cubes, and there is an infinite family of such puzzles. The chapter's primary argument is that, for any odd positive integer n = 2k + 1, there is exactly one way, up to symmetry, to make an n × n × n cube out of n tiny 1 × 1 × 1 cubes and six of each of a set of rectangular blocks. The chapter describes a way to solve each puzzle in the family and explains why there are no other solutions. It then presents several related open problems.

Agents and Implications of Foreign Land Deals in East African Community

2015 ◽  
pp. 263-286 ◽  
Author(s):  
Evans S. Osabuohien ◽  
Ciliaka M. Gitau ◽  
Uchenna R. Efobi ◽  
Michael Bruentrup
Keyword(s):  
Interesting Case ◽  
East African ◽  
Community Factors ◽  
East African Community ◽  
Foreign Land ◽  
African Community ◽  
One Step ◽  
Main Determinants ◽  
Global Increase ◽  
Community Outcome

Some of the factors that have been attributed to the global increase of Foreign Land Deals (FLDs) include the three Fs (food, fuel, and finance) crises, among others. However, most of the empirical evidence stems from the assessment of a broad set of countries. An analysis on the main determinants across host communities within a country presents specificity and closer reality. This chapter contributes by examining the community factors that could exert significant influence on determining whether or not a community receives FLDs in East African Community (EAC), focusing on Uganda. Uganda is an interesting case to investigate because the country is one of the destinations of FLDs in EAC, apart from Kenya and Tanzania. Taking it one step further, the chapter investigates the possible implications of FLDs on the host communities in terms of improvement (or deterioration) on selected community outcome variables: the quality and services relating to education, road, water, and health facilities.

Direct Product of Derived Steiner Systems Using Inversive Planes

1981 ◽  
Vol 33 (6) ◽  
pp. 1365-1369 ◽  
Author(s):  
K. T. Phelps
Keyword(s):  
Direct Product ◽  
Triple System ◽  
Steiner Triple System ◽  
Steiner System ◽  
Steiner Systems

A Steiner system S(t, k, v) is a pair (P, B) where P is a v-set and B is a collection of k-subsets of P (usually called blocks) such that every t-subset of P is contained in exactly one block of B. As is well known, associated with each point x ∈ P is a S(t � 1, k � 1, v � 1) defined on the set Px = P\{x} with blocksB(x) = {b\{x}|x ∈ b and b ∈ B}.The Steiner system (Px, B(x)) is said to be derived from (P, B) and is called (obviously) a derived Steiner (t – 1, k – 1)-system. Very little is known about derived Steiner systems despite much effort (cf. [11]). It is not even known whether every Steiner triple system is derived.Steiner systems are closely connected to equational classes of algebras (see [7]) for certain values of k.

scholarly journals EXISTENCE OF -ANALOGS OF STEINER SYSTEMS

2016 ◽  
Vol 4 ◽  
Author(s):  
MICHAEL BRAUN ◽  
TUVI ETZION ◽  
PATRIC R. J. ÖSTERGÅRD ◽  
ALEXANDER VARDY ◽  
ALFRED WASSERMANN
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Automorphism Groups ◽  
Steiner System ◽  
Dimensional Subspace ◽  
Steiner Systems ◽  
Cover Problem ◽  
Exact Cover

Let $\mathbb{F}_{q}^{n}$ be a vector space of dimension $n$ over the finite field $\mathbb{F}_{q}$. A $q$-analog of a Steiner system (also known as a $q$-Steiner system), denoted ${\mathcal{S}}_{q}(t,\!k,\!n)$, is a set ${\mathcal{S}}$ of $k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$ such that each $t$-dimensional subspace of $\mathbb{F}_{q}^{n}$ is contained in exactly one element of ${\mathcal{S}}$. Presently, $q$-Steiner systems are known only for $t\,=\,1\!$, and in the trivial cases $t\,=\,k$ and $k\,=\,n$. In this paper, the first nontrivial $q$-Steiner systems with $t\,\geqslant \,2$ are constructed. Specifically, several nonisomorphic $q$-Steiner systems ${\mathcal{S}}_{2}(2,3,13)$ are found by requiring that their automorphism groups contain the normalizer of a Singer subgroup of $\text{GL}(13,2)$. This approach leads to an instance of the exact cover problem, which turns out to have many solutions.

scholarly journals Ramsey numbers for certain k-graphs

1981 ◽  
Vol 30 (3) ◽  
pp. 284-296 ◽  
Author(s):  
Stefan A. Burr ◽  
Richard A. Duke
Keyword(s):  
Lower Bounds ◽  
Complete Solution ◽  
Ramsey Number ◽  
Upper And Lower Bounds ◽  
Uniform Hypergraph ◽  
Steiner Systems ◽  
Ramsey Numbers ◽  
Existence Question

AbstractWe are interested here in the Ramsey number r(T, C), where C is a complete k-uniform hypergraph and T is a “tree-like” k-graph. Upper and lower bounds are found for these numbers which lead, in some cases, to the exact value for r(T, C) and to a generalization of a theorem of Chváta1 on Ramsey numbers for graphs. In other cases we show that a determination of the exact values of r(T, C) would be equivalent to obtaining a complete solution to existence question for a certain class of Steiner systems.

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