Matroids You Have Known

Neel, David L.; Neudauer, Nancy Ann

doi:10.1080/0025570x.2009.11953589

Cited by 12 publications

(7 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proposition 2.4 establishes that every cycle system is a matroid. There is an abundance of literature, of which [9,20,21] are only representative. More relevent is [14] which emphasises independence, and dependence.…”

Section: Basic Cyclesmentioning

confidence: 99%

See 1 more Smart Citation

Cycle Systems

Pfaltz¹

2020

Math. Appl.

View full text Add to dashboard Cite

In this paper we show that the composition (symmetric difference) of cycles is well-defined. So, such a collection {.. . , C i , C k ,. . .} of cycles with a composition operator, • , is a matroid. As such, it has sets of independent, or basis, cycles that determine its rank r. This paper is concerned with independent and dependent sets of cycles within a cycle system. In particular, we enumerate the number of all possible basis sets in any cycle system of rank r ≤ 6. Then we use a generating function to establish that the ratio of basis sets to all possible r element sets approaches c, 0.287 < c < 0.289.

show abstract

Section: Basic Cyclesmentioning

confidence: 99%

“…Since all bases (maximal independent sets) have the same cardinality r (rank), the set of edges can be called a "graphic matroid". It is simplest possible example of the matroid concept and is, thus, found in many texts [9,20]. This model is utterly clear.…”

Section: Introductionmentioning

confidence: 98%

Cycle Systems

Pfaltz¹

2020

Math. Appl.

View full text Add to dashboard Cite

show abstract

“…Neither of the parallel connection and the series connection distributes over each other. 9 The greedy algorithm There exists a well-known characterization of matroids which, intriguingly, is algorithmic in nature and exemplifies the connection between matroids and problems in combinatorics [21,20]. The optimization problem for the pair (I, w) is to find a maximal member B of I of maximum weight.…”

Section: Constructionsmentioning

confidence: 99%

The Category of Matroids

Heunen

Patta

2017

Appl Categor Struct

View full text Add to dashboard Cite

The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and vector spaces, the latter being faithful and having a nearly full Kan extension; there is a functor to the category of geometric lattices, that is nearly full; there are various adjunctions and free constructions on subcategories, inducing a simplification monad; there are two orthogonal factorization systems; some, but not many, combinatorial constructions from matroid theory are functorial. Finally, a characterization of matroids in terms of optimality of the greedy algorithm can be rephrased in terms of limits.

show abstract

“…Hence the description with matroids will be very general (and quite pooras Whitney: "The fundamental question of completely characterizing systems which represent matrices is left unsolved") but it has the great advantage of onlycharacterizing the relationship between elements in two modes: independence and dependence. And this remains valid for a large set of objects -Whitney: "In place of a matrix we may equally well consider points or vectors in a Euclidean space, or polynomials, etc…" and more recently: graphs, matrices, groups, algebraic extensions… (for a pedagogical introduction to matroid, see (Neel and Neudauer 2009)).…”

Section: Endogenous Dynamics Of Technologies : the Design Of Interdepmentioning

confidence: 99%

Designing techniques for systemic impact: lessons from C-K theory and matroid structures

et al. 2016

View full text Add to dashboard Cite

International audienceAs underlined in Arthur " s book " the nature of technology " we are very knowledgeable on the design of objects, services or technical systems, but we don " t know much on the dynamics of technologies. Still contemporary innovation often consists in designing techniques with systemic impact. They are pervasive– both invasive and perturbing-, they recompose the family of techniques. Can we model the impact and the design of such techniques? More specifically: how can one design generic technology, ie a single technology that provokes a complete reordering of families of techniques? Recent advances in design theories open new possibilities to answer these questions. In this paper we use C-K design theory and a matroid-based model of the set of techniques to propose a new model (C-K/Ma) of the dynamics of techniques, accounting for the design of generic technologies. We show:  F1: C-K/Ma offers a computational model for designing a technique with systemic impact. This model sheds new light on the logics of combination in design – the model helps to identify four logics of " combinations " , from non-generative combinatorics to generative one

show abstract

Matroids You Have Known

Cited by 12 publications

References 6 publications

Cycle Systems

Cycle Systems

The Category of Matroids

Designing techniques for systemic impact: lessons from C-K theory and matroid structures

Contact Info

Product

Resources

About