A geometric permutation induced by a transversal line of a finite family F of disjoint convex sets in Rd is the order in which the transversal meets the members of the family. We prove that for each natural k, each family of k permutations is realizable (as a family of geometric permutations of some F ) in Rd for d ≥ 2k − 1, but there is a family of k permutations which is non-realizable in Rd ...

We will present that the forbidden minors for 3-realizable graph are K5 and the octahedron. You can directly go to Theorem 13 and if necessary refer to the definitions, lemmas and theorems before it. Definition 1 A graph G is d-realizable if, given any realization p1, · · · , pn of the graph in some finite dimensional Euclidean space, there exists a realization q1, · · · , qn in E with the same...

For each positive integer n ≥ 4, we give an inequality satisfied by rank functions of arrangements of n subspaces. When n = 4 we recover Ingleton’s inequality; for higher n the inequalities are all new. These inequalities can be thought of as a hierarchy of necessary conditions for a (poly)matroid to be realizable. Some related open questions about the “cone of realizable polymatroids” are also...

Linear elasticity for a ber reinforced two-component periodic composite is treated numerically. Given volume fractions and elastic moduli of the two components, which are the realizable combinations of the eeective moduli? We solve a reduced problem for a system of concentric cylindrical shells arranged in a hexagonal unit cell. The eeective moduli are estimated via third-order bounds, which in...

Congruent coordinate transformations are used to convert second-order models to a form in which the mass, damping and stiffness matrices can be interpreted as a passive mechanical system. For those systems which can be constructed from interconnected mass, stiffness and damping elements, it is shown that the input-output preserving transformations can be parameterized by an orthogonal matrix wh...

A tournament T,, consists of a set of n vertices and a single directed edge joining every pair of distinct vertices. We denote the vertices of T’. by {l, . . . , n). A permutation aI, . . . , a, of the vertices is a generalized path. The type af the path is characterized by the sequence a,_1 = e3 l l l E~__~ where si = i-1 if e -+ ai+1 and 6 = -1 if a, -+ ai tl. We also say that mn_l is realize...

In [13], Kleene formulated a truth-notion called "readability" for formulas of intuitionistic number theory(1). David Nelson(2) showed that every number-theoretic formula deducible in the intuitionistic predicate calculus(3) (stated by means of schemata, without proposition or predicate variables) from realizable number-theoretic formulas is realizable. In particular, then, every formula in the...

Below we discuss the partition of the space of real univariate polynomials according to the number of positive and negative roots and signs of the coefficients. We present several series of non-realizable combinations of signs together with the numbers of positive and negative roots. We provide a detailed information about possible non-realizable combinations up to degree 8 as well as a general...