Fano Matroid Research Articles

THE USE OF MARKOV ALGORITHMS FOR THE STUDY OF l-VOIDS IN BIG DATA OF SOCIO-ECONOMIC SYSTEMS. PART 2

The second part of the article is presented. Big data l-voids are considered using the N-scheme of the Markov algorithm. The diagrams of occurrences of l-voids in a semi-Eulerian cycle containing an Euler path, a matroid and an incomplete Fano matroid, minors K3, 3 and K5, an extra large cycle of occurrences are analyzed. An example of reconstructing a fragment of an incomplete Fano matroid with l-voids is considered. Examples are given for independent implementation of the method of filling the artificial intelligence database (AnwM f typeK) DB based on the results of the analysis of l-voids in the N-scheme of the Markov algorithm. The database is populated using a formal language with the alphabet M = litj/abdgckm where litj are terminal elements, abdgckm are non-terminal elements, and the basis of the alphabet is i – initial, t – terminal, j – isomorphic, l-empty morphisms. The p – initial, e – simple, and h – final occurrences of words from the Markov alphabet A are determined. An example of a simplified numbering of occurrence chains is given, which ensures the selection of all objects represented by words in the Markov alphabet A, belonging to “their own” chain of occurrences, as well as all related to chain object. The mechanism of dynamic change of the number-type in the N-scheme of the Markov algorithm is presented, which provides the analysis of the word as an evolving category, as well as the evolution of categories, which includes such a word. It has been established that normal Markov inference is characterized by temporal conditionality and locality. The conclusion is made about the expediency of considering the N-scheme in a distributed computing system. In this case, it is necessary to analyze the diagrams of occurrences of objects that have several designations with words from the Markov alphabet A, in order to identify them and process data from such objects.

On the reliability roots of simplicial complexes and matroids

Assume that the vertices of a graph G are always operational, but the edges of G fail independently with probability q∈[0,1]. The all-terminal reliability of G is the probability that the resulting subgraph is connected. The all-terminal reliability can be formulated into a polynomial in q, and it was conjectured that all the roots of (nonzero) reliability polynomials fall inside the closed unit disk. It has since been shown that there exist some connected graphs which have their reliability roots outside the closed unit disk, but these examples seem to be few and far between, and the roots are only barely outside the disk. In this paper we generalize the notion of reliability to simplicial complexes and matroids and investigate when the roots fall inside the closed unit disk. We show that such is the case for matroids of rank 3 and paving matroids of rank 4. We also prove that the reliability roots of shellable complexes are dense in the complex plane, and that the real reliability roots of any matroid lie in [−1,0)∪{1}. Finally, we also show that the reliability roots of thickenings of the Fano matroid can lie outside the unit disk.

Quasiregular Matroids

Regular matroids are binary matroids with no minors isomorphic to the Fano matroid $F_7$ or its dual $F_7^*$. Seymour proved that 3-connected regular matroids are either graphs, cographs, or $R_{10}$, or else can be decomposed along a non-minimal exact 3-separation induced by $R_{12}$. Quasiregular matroids are binary matroids with no minor isomorphic to the self-dual binary matroid $E_4$. The class of quasiregular matroids properly contains the class of regular matroids. We prove that 3-connected quasiregular matroids are either graphs, cographs, or deletion-minors of $PG(3,2)$, $R_{17}$ or $M_{12}$ or else can be decomposed along a non-minimal exact 3-separation induced by $R_{12}$, $P_9$, or $P_9^*$.

Characterizing binary matroids with no [formula omitted]-minor

In this paper, we give a complete characterization of binary matroids with no P9-minor. A 3-connected binary matroid M has no P9-minor if and only if M is a 3-connected regular matroid, a binary spike with rank at least four, one of the internally 4-connected non-regular minors of a special 16-element matroid Y16, or a matroid obtained by 3-summing copies of the Fano matroid to a 3-connected cographic matroid M⁎(K3,n), M⁎(K3,n′), M⁎(K3,n″), or M⁎(K3,n‴) (n≥2). Here the simple graphs K3,n′, K3,n″, and K3,n‴ are obtained from K3,n by adding one, two, or three edges in the color class of size three, respectively.

Minimal non-orientable matroids of rank three

Minimal non-orientable matroids have been investigated as a threshold between orientability and non-orientability. The Fano matroid and the MacLane matroid are well-known minimal non-orientable matroids of rank 3. A natural question is whether there exists a minimal non-orientable matroid of every rank r with m elements. In this paper, we give an answer to the question in rank 3 that for every m≥7, there exists a minimal non-orientable matroid of rank 3 with m elements. To prove this statement, we construct two new infinite families of minimal non-orientable matroids of rank 3. Furthermore we also investigate representability of the two infinite families.

Matroid Secretary for Regular and Decomposable Matroids

In the matroid secretary problem we are given a stream of elements in random order and asked to choose a set of elements that maximizes the total value of the set, subject to being an independent set of a matroid given in advance. The difficulty comes from the assumption that decisions are irrevocable: if we choose to accept an element when it is presented by the stream, then we can never get rid of it, and if we choose not to accept it, then we cannot later add it. Babaioff, Immorlica, and Kleinberg [Proceedings of SODA'07] introduced this problem, gave O(1)-competitive algorithms for certain classes of matroids, and conjectured that every matroid admits an O(1)-competitive algorithm. However, most matroids that are known to admit an O(1)-competitive algorithm can be easily represented using graphs (e.g., graphic, cographic, and transversal matroids). In particular, there is very little known about $F$-representable matroids (the class of matroids that can be represented as elements of a vector space over a field $F$), which are one of the foundational types of matroids. Moreover, most of the known techniques are as dependent on graph theory as they are on matroid theory. We go beyond graphs by giving O(1)-competitive algorithms for regular matroids (the class of matroids that are representable over any field) and use techniques that are fundamentally matroid-theoretic rather than graph-theoretic. Our main technique is to leverage the seminal regular matroid decomposition theorem of Seymour, which gives a method for decomposing any regular matroid into matroids which are graphic, cographic, or isomorphic to a simple 10-element matroid. We show how to combine in a black-box manner any algorithms for these basic classes into an algorithm for a given regular matroid, i.e., how to respect the decomposition. In fact, this allows us to generalize beyond regular matroids to any class of matroids that admits such a decomposition into classes for which we already have good algorithms. In particular, we give an O(1)-competitive algorithm for the class of max-flow min-cut matroids, which Seymour showed can be decomposed into regular matroids and copies of the Fano matroid.

Some exceptional cases in mathematics: euler characteristic, division algebras, cross vector product and fano matroid

We review remarkable results in several mathematical scenarios, including graph theory, division algebras, cross product formalism and matroid theory. Specifically, we mention the following subjects: (1) the Euler relation in graph theory, and its higher-dimensional generalization, (2) the dimensional theorem for division algebras and in particular the Hurwitz theorem for normed division algebras, (3) the vector cross product dimensional possibilities, (4) some theorems for graphs and matroids. Our main goal is to motivate a possible research work in these four topics, putting special interest in their possible links

A Characterization of Box-Mengerian Matroid Ports

Let M be a matroid on E∪{l}, where l ∉ E is a distinguished element of M. The l-port of M is the set 𝒫= {P: P ⊆ E with P∪{l} a circuit of M}. Let A be the 𝒫-E incidence matrix. Let U2, 4 be the uniform matroid on four elements of rank two, let F7 be the Fano matroid, let F7* be the dual of F7, and let F7+ be the unique series extension of F7. In this paper, we prove that the system Ax≥1, x≥0 is box-totally dual integral (box-TDI) if and only if M has no U2, 4-minor using l, no F7*-minor using l, and no F7+-minor using l as a series element. Our characterization yields a number of interesting results in combinatorial optimization.

Polynomials with the half-plane property and matroid theory

A polynomial f is said to have the half-plane property if there is an open half-plane H ⊂ C , whose boundary contains the origin, such that f is non-zero whenever all the variables are in H. This paper answers several open questions relating multivariate polynomials with the half-plane property to matroid theory. (1) We prove that the support of a multivariate polynomial with the half-plane property is a jump system. This answers an open question posed by Choe, Oxley, Sokal and Wagner and generalizes their recent result claiming that the same is true whenever the polynomial is also homogeneous. (2) We prove that a multivariate multi-affine polynomial f ∈ R [ z 1 , … , z n ] has the half-plane property (with respect to the upper half-plane) if and only if ∂ f ∂ z i ( x ) ⋅ ∂ f ∂ z j ( x ) − ∂ 2 f ∂ z i ∂ z j ( x ) ⋅ f ( x ) ⩾ 0 for all x ∈ R n and 1 ⩽ i , j ⩽ n . This is used to answer two open questions posed by Choe and Wagner regarding strongly Rayleigh matroids. (3) We prove that the Fano matroid is not the support of a polynomial with the half-plane property. This is the first instance of a matroid which does not appear as the support of a polynomial with the half-plane property and answers a question posed by Choe et al. We also discuss further directions and open problems.

SEARCH FOR A "GRAVITOID" THEORY

By combining the concepts of graviton and matroid, we outline a new gravitational theory which we call gravitoid theory. The idea of this theory emerged as an attempt to link the mathematical structure of matroid theory with M theory. Our observations are essentially based on the formulation of matroid bundle due to MacPherson and Anderson–Davis. Also, by considering the oriented matroid theory, we add new observations about the link between the Fano matroid and D=11 supergravity which was discussed in some of our recent papers. In particular we find a connection between the affine matroid AG(3, 2) and the G2-symmetry of D=11 supergravity.

Matroid theory and Chern–Simons

It is shown that matroid theory may provide a natural mathematical framework for a duality symmetries not only for quantum Yang–Mills physics, but also for M-theory. Our discussion is focused in an action consisting purely of the Chern–Simons term, but in principle the main ideas can be applied beyond such an action. In our treatment the theorem due to Thistlethwaite, which gives a relationship between the Tutte polynomial for graphs and Jones polynomial for alternating knots and links, plays a central role. Before addressing this question we briefly mention some important aspects of matroid theory and we point out a connection between the Fano matroid and D=11 supergravity. Our approach also seems to be related to loop solutions of quantum gravity based in an Ashtekar formalism.

Large Circuits in Binary Matroids of Large Cogirth, I

Let F 7 denote the Fano matroid and e be a fixed element of F 7 . Let P ( F 7 , e ) be the family of matroids obtained by taking the parallel connection of one or more copies of F 7 about e . Let M be a simple binary matroid such that every cocircuit of M has size at least d ⩾3. We show that if M does not have an F 7 -minor, M ≠ F * 7 , and d ⩾( r ( M )+1)/2 then M has a circuit of size r ( M )+1. We also show that if M is connected, e ∈ E ( M ), M does not have both an F 7 -minor and an F * 7 -minor, and M ∉ P ( F 7 , e ), then M has a circuit that contains e and has size at least d +1.

Large Circuits in Binary Matroids of Large Cogirth, II

Let F 7 denote the Fano matroid and M be a simple connected binary matroid such that every cocircuit of M has size at least d ⩾3. We show that if M does not have an F 7 -minor, M ≠ F * 7 , and d ∉{5, 6, 7, 8}, then M has a circuit of size at least min{ r ( M )+1, 2 d }. We conjecture that the latter result holds for all d ⩾3.

A Characterization of Box 1d-Integral Binary Clutters

Let Q6 denote the port of the dual Fano matroid F*7 and let Q7 denote the clutter consisting of the circuits of the Fano matroid F7 that contain a given element. Let L be a binary clutter on E and let d ≥ 2 be an integer. We prove that all the vertices of the polytope {x ∈ RE+ | x(C) ≥ 1 for C ∈ L} ∩ {x | a ≤ x ≤ b} are 1d-integral, for any 1d-integral a, b, if and only if L does not have Q6 or Q7 as a minor. This includes the class of ports of regular matroids. Applications to graphs are presented, extending a result from Laurent and Pojiak [7].

On sign-representable matroids

AbstractA matroid M will be called sign-representable if, for every basis B of M, there is a (0, 1, −l)-matrix [Ir|y] representing M over ℚ in which the first r columns correspond to the members of B. The class of sign-representable matroids, which is closely related to the important class of regular matroids, is easily seen to be closed under both duality and the taking of minors. This paper proves several characterizations of the class, including a constructive one, and shows that the excluded minors for the class are U2, 5, U3, 5, the Fano matroid and its dual, and the rank-3 whirl.

A decomposition of the matroids with the max-flow min-cut property

A matroid has the max-flow min-cut property if a certain circuit packing problem has an optimal solution that is integral whenever the capacities assigned to the elements of the matroid are integral. P.D. Seymour characterized the matroids with this property in terms of minimal forbidden minors. Here we employ a variant of a previously developed decomposition algorithm to produce two decomposition theorems for this matroid class. The first theorem roughly says that any 3-connected matroid of the class is regular, or equal to the Fano matroid, or is a 3-sum. The second theorem is quite similar, but involves a more detailed analysis of the 3-sum case and includes an additional case.

On the algebraic characteristic set for a class of matroids

The independent sets of an algebraic matroid are sets of algebraically independent transcendentals over a field k k . If a matroid M M is isomorphic to an algebraic matroid the latter is called an algebraic representation of M M . Vector representations of matroids are defined similarly. A matroid may have algebraic (resp. vector) representations over fields of different characteristics. The problem in which characteristic sets are possible for vector representations was recently answered (see [2]). The corresponding problem for algebraic representations is open. We consider a class of matroids M p {M_p} ( p p a prime) the vector representations which were determined by T. Lazarson long ago. One member of this class, M 2 {M_2} , is the important Fano matroid which plays a crucial role in many parts of matroid theory. We prove that M p {M_p} has algebraic representations only over fields of characteristic p p . The proof depends on derivations in fields. Using derivations we transform an algebraic representation of M p {M_p} into a vector representation.

A note on nongraphic matroids

Tutte found an excluded minor characterization of graphic matroids with five excluded minors. A variation on Tutte's result is presented here. Let { e, f, g} be a circuit of a 3-connected nongraphic matroid M. Then M has a minor using e, f, g isomorphic to either the 4-point line, the Fano matroid, or the bond matroid of K 3,3.

Matroid representation over GF(3)

We show that a matroid is representable over GF(3) if and only if no minor is the five-point line or the Fano matroid, or their duals. Tutte's famous characterization of the regular matroids is a corollary. A key lemma states that two representations of the same matroid in the same vector space over GF(3) may be transformed one into the other by inverting some points through the origin and taking a linear transformation; no result of this kind holds in larger fields.

On Reid's characterization of the ternary matroids

In this paper we prove a stronger version of a result of Ralph Reid characterizing the ternary matroids (i.e., the matroids representable over the field of 3 elements, GF(3)). In particular, we prove that a matroid is ternary if it has no seriesminor of type L n for n ≥ 5 ( n cells and n circuits, each of size n − 1), and no series-minor of type L 5 ∗ (dual of L 5), BII (Fano matroid) or BI (dual of type BII). The proof we give does not assume Reid's theorem. Rather we give a direct proof based on the methods (notably the homotopy theorem) developed by Tutte for proving his characterization of regular matroids. Indeed, the steps involved in our proof closely parallel Tutte's proof, but carrying out these steps now becomes much more complicated.