scholarly journals Steiner systems and configurations of points

2020 ◽  
Author(s):  
Edoardo Ballico ◽  
Giuseppe Favacchio ◽  
Elena Guardo ◽  
Lorenzo Milazzo
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Design Theory ◽  
Hilbert Function ◽  
Linear Codes ◽  
Betti Numbers ◽  
Steiner System ◽  
Steiner Systems ◽  
Waldschmidt Constant ◽  
The Ideal

Abstract The aim of this paper is to make a connection between design theory and algebraic geometry/commutative algebra. In particular, given any Steiner System S(t, n, v) we associate two ideals, in a suitable polynomial ring, defining a Steiner configuration of points and its Complement. We focus on the latter, studying its homological invariants, such as Hilbert Function and Betti numbers. We also study symbolic and regular powers associated to the ideal defining a Complement of a Steiner configuration of points, finding its Waldschmidt constant, regularity, bounds on its resurgence and asymptotic resurgence. We also compute the parameters of linear codes associated to any Steiner configuration of points and its Complement.

scholarly journals Breakout Group Allocation Schedules and the Social Golfer Problem with Adjacent Group Sizes

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 13
Author(s):  
Alice Miller ◽  
Matthew Barr ◽  
William Kavanagh ◽  
Ivaylo Valkov ◽  
Helen C. Purchase
Keyword(s):  
Design Theory ◽  
Teaching Experience ◽  
Combinatorial Design ◽  
Breakout Group ◽  
Online Resource ◽  
The Social ◽  
Adjacent Group ◽  
Group Allocation ◽  
Novel Variation ◽  
The Ideal

The current pandemic has led schools and universities to turn to online meeting software solutions such as Zoom and Microsoft Teams. The teaching experience can be enhanced via the use of breakout rooms for small group interaction. Over the course of a class (or over several classes), the class will be allocated to breakout groups multiple times over several rounds. It is desirable to mix the groups as much as possible, the ideal being that no two students appear in the same group in more than one round. In this paper, we discuss how the problem of scheduling balanced allocations of students to sequential breakout rooms directly corresponds to a novel variation of a well-known problem in combinatorics (the social golfer problem), which we call the social golfer problem with adjacent group sizes. We explain how solutions to this problem can be obtained using constructions from combinatorial design theory and how they can be used to obtain good, balanced breakout room allocation schedules. We present our solutions for up to 50 students and introduce an online resource that educators can access to immediately generate suitable allocation schedules.

On Grothendieck's local duality theorem

1979 ◽  
Vol 85 (3) ◽  
pp. 431-437 ◽  
Author(s):  
M. H. Bijan-Zadeh ◽  
R. Y. Sharp
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Noetherian Ring ◽  
Duality Theorem ◽  
Great Emphasis ◽  
Triangulated Category ◽  
Elementary Approach ◽  
Commutative Noetherian Ring ◽  
Local Duality ◽  
Dualizing Complexes

In (11) and (12), a comparatively elementary approach to the use of dualizing complexes in commutative algebra has been developed. Dualizing complexes were introduced by Grothendieck and Hartshorne in (2) for use in algebraic geometry; the approach to dualizing complexes in (11) and (12) differs from that of Grothendieck and Hartshorne in that it avoids use of the concepts of triangulated category, derived category, and localization of categories, and instead places great emphasis on the concept of quasi-isomorphism of complexes of modules over a commutative Noetherian ring.

Cellular reductions of the Pommaret-Seiler resolution for Quasi-stable ideals

2021 ◽  
Vol 55 (3) ◽  
pp. 102-106
Author(s):  
Rodrigo Iglesias ◽  
Eduardo Sáenz de Cabezón
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Combinatorial Properties ◽  
Pommaret Bases ◽  
Involutive Bases

Involutive bases were introduced in [6] as a type of Gröbner bases with additional combinatorial properties. Pommaret bases are a particular kind of involutive bases with strong relations to commutative algebra and algebraic geometry[11, 12].

The ideal entropy of BCI-algebras and its application in the binary linear codes

2018 ◽  
Vol 23 (1) ◽  
pp. 39-57
Author(s):  
Mohammad Ebrahimi ◽  
Alireza Izadara
Keyword(s):  
Linear Codes ◽  
Binary Linear Codes ◽  
The Ideal

scholarly journals Bundles on with vanishing lower cohomologies

2020 ◽  
pp. 1-20
Author(s):  
Mengyuan Zhang
Keyword(s):  
Hilbert Function ◽  
Betti Numbers ◽  
Projective Spaces ◽  
Hilbert Functions ◽  
Free Resolutions ◽  
Rational Varieties ◽  
Minimal Free Resolutions ◽  
Graded Lattice

Abstract We study bundles on projective spaces that have vanishing lower cohomologies using their short minimal free resolutions. We partition the moduli $\mathcal{M}$ according to the Hilbert function H and classify all possible Hilbert functions H of such bundles. For each H, we describe a stratification of $\mathcal{M}_H$ by quotients of rational varieties. We show that the closed strata form a graded lattice given by the Betti numbers.

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.

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