Golden-Mean and Secret Sharing Matroids

<p>Maximum-sized results are an important part of matroid theory, and results currently exist for various classes of matroids. Archer conjectured that the maximum-sized golden-mean matroids fall into three distinct classes, as op- posed to the one class of all current results. We will prove a partial result that we hope will lead to a full proof. In the second part of this thesis, we look at secret sharing matroids, with a particular emphasis on the class of group-induced p-representable matroids, as introduced by Matúš. We give new proofs for results of Matúš', relating to M(K₄), F₇ and F⁻₇. We show that the techniques used do not extend in some natural ways, and pose some unanswered questions relating to the structure of secret sharing matroids.</p>

Golden-Mean and Secret Sharing Matroids

<p>Maximum-sized results are an important part of matroid theory, and results currently exist for various classes of matroids. Archer conjectured that the maximum-sized golden-mean matroids fall into three distinct classes, as op- posed to the one class of all current results. We will prove a partial result that we hope will lead to a full proof. In the second part of this thesis, we look at secret sharing matroids, with a particular emphasis on the class of group-induced p-representable matroids, as introduced by Matúš. We give new proofs for results of Matúš', relating to M(K₄), F₇ and F⁻₇. We show that the techniques used do not extend in some natural ways, and pose some unanswered questions relating to the structure of secret sharing matroids.</p>

Sublinear Circuits for Polyhedral Sets

Sublinear circuits are generalizations of the affine circuits in matroid theory, and they arise as the convex-combinatorial core underlying constrained non-negativity certificates of exponential sums and of polynomials based on the arithmetic-geometric inequality. Here, we study the polyhedral combinatorics of sublinear circuits for polyhedral constraint sets. We give results on the relation between the sublinear circuits and their supports and provide necessary as well as sufficient criteria for sublinear circuits. Based on these characterizations, we provide some explicit results and enumerations for two prominent polyhedral cases, namely the non-negative orthant and the cube [− 1,1]n.

A New Description of Transversal Matroids Through Rough Set Approach

Matroid theory is a useful tool for the combinatorial optimization issue in data mining, machine learning and knowledge discovery. Recently, combining matroid theory with rough sets is becoming interesting. In this paper, rough set approaches are used to investigate an important class of matroids, transversal matroids. We first extend the concept of upper approximation number functions in rough set theory and propose the notion of generalized upper approximation number functions on a set system. By means of the new notion, we give some necessary and sufficient conditions for a subset to be a partial transversal of a set system. Furthermore, we obtain a new description of a transversal matroid by the generalized upper approximation number function. We show that a transversal matroid can be induced by the generalized upper approximation number function which can be decomposed into the sum of some elementary generalized upper approximation number functions. Conversely, we also prove that a generalized upper approximation number function can induce a transversal matroid. Finally, we apply the generalized upper approximation number function to study the relationship among transversal matroids.

Matroidal Entropy Functions: A Quartet of Theories of Information, Matroid, Design and Coding

In this paper, we study the entropy functions on extreme rays of the polymatroidal region which contain a matroid, i.e., matroidal entropy functions. We introduce variable strength orthogonal arrays indexed by a connected matroid M and positive integer v which can be regarded as expanding the classic combinatorial structure orthogonal arrays. It is interesting that they are equivalent to the partition-representations of the matroid M with degree v and the (M,v) almost affine codes. Thus, a synergy among four fields, i.e., information theory, matroid theory, combinatorial design, and coding theory is developed, which may lead to potential applications in information problems such as network coding and secret-sharing. Leveraging the construction of variable strength orthogonal arrays, we characterize all matroidal entropy functions of order n≤5 with the exception of log10·U2,5 and logv·U3,5 for some v.

Can the Genericity Assumption Decrease the Rank of a Matrix?

The genericity assumption, supposing that the nonzero parameters of a system are algebraically independent transcendentals over the field of the rationals, often helps for the mathematical modelling of linear systems. Without this condition nonzero expansion members of a determinant can cancel out each other, decreasing the rank of a matrix. In this note we show that under some circumstances an increase is also possible. This counterintuitive phenomenon is explained using some tools from matroid theory, and is illustrated by a classical network of Carlin and Youla.

Reconciling enumeration contradictions: Complete list of Baranov chains with up to 15 links with mathematical proof

The identification of Baranov chains is associated with the rigid subchain identification problem, which is a crucial step in several methods of structural synthesis of kinematic chains. In this paper, a systematic approach for the detection of rigid subchains, based on matroid theory, is presented and proved. Based on this approach, a novel method for the enumeration of Baranov chains is proposed. A novel algorithm is applied to a database of non-isomorphic graphs of non-fractionated zero-mobility kinematic chains. By means of the proposed algorithm, the previous results for Baranov chains presented in literature with up to 11 links are compared and validated. Furthermore, discrepancies in the number of Baranov chains with up to 13 links, presented in literature, are pointed out, discussed and the proven results are presented. Finally, the complete family of Baranov chains with up to 15 links is obtained. Examples of application of the proposed method are provided.

Epsilon local rigidity and numerical algebraic geometry

A well-known combinatorial algorithm can decide generic rigidity in the plane by determining if the graph is of Pollaczek–Geiringer–Laman type. Methods from matroid theory have been used to prove other interesting results, again under the assumption of generic configurations. However, configurations arising in applications may not be generic. We present Theorem 4.2 and its corresponding Algorithm 1 which decide if a configuration is [Formula: see text]-locally rigid, a notion we define. A configuration which is [Formula: see text]-locally rigid may be locally rigid or flexible, but any continuous deformations remain within a sphere of radius [Formula: see text] in configuration space. Deciding [Formula: see text]-local rigidity is possible for configurations which are smooth or singular, generic or non-generic. We also present Algorithms 2 and 3 which use numerical algebraic geometry to compute a discrete-time sample of a continuous flex, providing useful visual information for the scientist.