We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. We show that a tropical oriented matroid determines a subdivision of a product of two simplices, and conjecture that this correspondence is a bijection.

A new matroid is introduced: this matroid is de0ned starting from any matroid and one of its bases, hence we call it base-matroid. Besides some properties of the base-matroid, a non-trivial algorithm for the solution of the related matroid optimization problem is presented. The new matroid has application in the 0eld of inverse combinatorial optimization problems. We discuss in detail the LP fo...

The mixed polynomial matrix, introduced as a convenient mathematical tool for the description of physical/engineering dynamical systems, is a polynomial matrix of which the coefficients are classified into fixed constants and independent parameters. The valuated matroid, invented by Dress and Wenzel [Appl. Math. Lett., 3 (1990), pp. 33–35], is a combinatorial abstraction of the degree of minors...

Lattice Subdivisions and Tropical Oriented Matroids Featuring ∆ n−1 × ∆ d−1 LindsayCPiechnik Subdivisions of products of simplices, and their applications, appear across mathematics. In this thesis, they are the tie between two branches of my research: polytopal lattice subdivisions and tropical oriented matroid theory. The first chapter describes desirable combinatorial properties of subdivisi...

We introduce “matroid parse trees” which, using only a limited amount of information at each node, can build up the vector representations of matroids of bounded branch-width over a finite field. We prove that if M is a family of matroids described by a sentence in the monadic second-order logic of matroids, then there is a finite tree automaton accepting exactly those parse trees which build v...

Johnsen, Roksvold and Verdure have shown how fundamental invariants concepts of linear coding theory, such as weight enumerators higher weights, can be studied in the context matroid theory. They that Betti numbers, a its elongations, associated to code, determine both enumerator weights matroid. In same vein, Britz, Mayhew Shiromoto, introduced demimatroids natural generalization matroids. are...

Let C2(M) be the second order circuit graph of a simple connected matroid M , then C2(M) is 2-connected if M has more than one circuit and M is not a line. Moreover, C2(M) has diameter at most two if and only if M does not have any restriction isomorphic to U2,6.

functional dependency structures Franti sek Mat u s Institute of Information Theory and Automation Czechoslovak Academy of Sciences Pod vodd arenskou v e z 4, 182 08 Prague Czechoslovakia Abstract The functional dependency (FD) structures known mainly from the database relationships theory are appeared to comprehend other kinds of dependencies among objects like functions, boolean or random var...

For an integer l ≥ 2, let U(l) be the class of matroids with no U2,l+2-minor. A matroid in U(l) is extremal if it is simple and has no simple rank-preserving singleelement extension in U(l). An amalgam of two matroids is a simultaneous extension of both on the union of the two ground sets. We study amalgams of extremal matroids in U(l): we determine which amalgams are in U(l) and which are extr...