On the representations of projective geometries in algebraic combinatorial geometries

In this paper, we show that the full algebraic combinatorial geometry is not a projective geometry, it is only semimodular, but the p-polynomial points give a projective subgeometry. Also, we show that the subgeometry can be coordinatized by a skew field, which is quotient ring of an Ore domain. As a corollary, we prove the existence of algebraic representations over fields of prime characteristic of the non-Pappus matroid and its dual matroid. Regarding the existence of algebraic representations of the non-Pappus matroid, this result was earlier proved by Lindström [7] for finite fields.     

An Affine Representation for Transversal Geometries

Pregeometries (matroids) whose independent sets are the partial matchings of a relation (transversal pregeometries) can be canonically imbedded in a free-simplicial pregeometry (one whose points lie freely on flats spanned by a simplex). Conversely, all subgeometries of such free-simplicial pregeometries are transversal. Free-simplicial pregeometries are counted and their duals are naturally constructed and shown to be free-simplicial (showing self-dual free-simplexes corrspond to quasisymmetric relations). For more general transversal pregeometries, modular flats are characterized and transversal contractions are exemplified. Binary transversal pregeometries and their contractions (the class of binary gammoids) are shown to be the class of series-parallel networks, providing insight for further characterizations of (coordinatized) gammoids by excluded minors. Theorem. All principal transversal pregeometries and their truncations have critical exponent at most 2.     

Some Results on Minimum Aberration Mixed-level （2^r）2^n Factorial Designs

Mukerjee and Wu(2001) employed projective geometry theory to find the wordlength pattern of a regular mixed factorial design in terms of its complementary set, but only for the numbers of words of length 3 or 4.In this paper,by introducing a concept of consulting design and based on the connection between factorial design theory and cod- ing theory,we obtain some combinatorial identities that relate the wordlength pattern of a regular mixed-level (2~r)2~n factorial design to that of its consulting design.Consequently,a general rule for identifying minimum aberration (2~r)2~n factorial designs through their con- sulting designs is established.It is an improvement and generalization of the related result in Mukerjee and Wu(2001).     

Integral geometry of complex space forms

We show how Alesker’s theory of valuations on manifolds gives rise to an algebraic picture of the integral geometry of any Riemannian isotropic space. We then apply this method to give a thorough account of the integral geometry of the complex space forms, i.e. complex projective space, complex hyperbolic space and complex Euclidean space. In particular, we compute the family of kinematic formulas for invariant valuations and invariant curvature measures in these spaces. In addition to new and more efficient framings of the tube formulas of Gray and the kinematic formulas of Shifrin, this approach yields a new formula expressing the volumes of the tubes about a totally real submanifold in terms of its intrinsic Riemannian structure. We also show by direct calculation that the Lipschitz-Killing valuations stabilize the subspace of invariant angular curvature measures, suggesting the possibility that a similar phenomenon holds for all Riemannian manifolds. We conclude with a number of open questions and conjectures.     

Minimum secondary aberration fractional factorial split-plot designs in terms of consulting designs

It is very powerful for constructing nearly saturated factorial designs to characterize fractional factorial (FF) designs through their consulting designs when the consulting designs are small. Mukerjee and Fang employed the projective geometry theory to find the secondary wordlength pattern of a regular symmetrical fractional factorial split-plot (FFSP) design in terms of its complementary subset, but not in a unified form. In this paper, based on the connection between factorial design theory and coding theory, we obtain some general and unified combinatorial identities that relate the secondary wordlength pattern of a regular symmetrical or mixed-level FFSP design to that of its consulting design. According to these identities, we further establish some general and unified rules for identifying minimum secondary aberration, symmetrical or mixed-level, FFSP designs through their consulting designs.     

Homomorphisms of Projective Remoteness Planes

A characterization of a class of homomorphisms of projective remoteness planes in terms of their coordinate rings is given. A remoteness preserving homomorphism of projective remoteness planes is factored into three homomorphisms of known types. Two of these are constructed from groups associated with the planes and the homomorphism that induces on the coordinate rings of the planes. The third is a covering of planes coordinatized by the same ring. This generalizes known results for projective planes, projective ring planes, and Moufang–Veldkamp planes.     

Polyhedral annexation in mixed integer and combinatorial programming

Polyhedral annexation is a new approach for generating all valid inequalities in mixed integer and combinatorial programming. These include the facets of the convex hull of feasible integer solutions. The approach is capable of exploiting the characteristics of the feasible solution space in regions both “adjacent to” and “distant from” the linear programming vertex without resorting to specialized notions of group theory, convex analysis or projective geometry. The approach also provides new ways for exploiting the “branching inequalities” of branch and bound.     

Diagrams,Tensors and Geometric Reasoning

Geometry and in particular projective geometry (and its corresponding invariant theory) deals a lot with structural properties of geometric objects and their interrelations. This papers describes how concepts of tensor calculus can be used to express geometric invariants and how, in particular, diagrammatic notation can be used to deal with invariants in a highly intuitive way. In particular we explain how geometries like euclidean or spherical geometry can be dealt with in this framework. Dedicated to the memory of Victor Klee, and in particular to his striving for conceptual simplicity     

Harmonic conjugation in harmonic matroids

We study a generalization of the concept of harmonic conjugation from projective geometry and full algebraic matroids to a larger class of matroids called harmonic matroids. We use harmonic conjugation to construct a projective plane of prime order in harmonic matroids without using the axioms of projective geometry. As a particular case we have a combinatorial construction of a projective plane of prime order in full algebraic matroids.     

On the smallest simple, unipotent Bol loop

Finite simple, unipotent Bol loops have recently been identified and constructed using group theory. However, the purely group-theoretical constructions of the actual loops are indirect, somewhat arbitrary in places, and rely on computer calculations to a certain extent. In the spirit of revisionism, this paper is intended to give a more explicit combinatorial specification of the smallest simple, unipotent Bol loop, making use of concepts from projective geometry and quasigroup theory along with the group-theoretical background. The loop has dual permutation representations on the projective line of order 5, with doubly stochastic action matrices.     

Projective Equivalence of Δ-Matroids with Coefficients and Symplectic Geometries

The projective equivalence of matroid representations over fields and of oriented matroids is well studied. This paper is devoted to the study of projective equivalence of Δ-matroids with coefficients, which covers the concept of projective equivalence of matroids with coefficients and thus in particular the projective equivalence of represented and oriented matroids. A necessary and sufficient condition for the projective equivalence of Δ-matroids with coefficients is established in terms of the inner Tutte group T _M ⁽⁰⁾ of the underlying combinatorial geometry M. The structure of T_M ⁽⁰⁾ of symplectic projective spaces M of odd dimensions d≥3 over fields is computed.     

Restricted homogeneous coordinates for the Cayley projective plane

I. Porteous has shown that the Cayley projective plane can be coordinatized in a way resembling homogeneous coordinates. We will show how to construct line coordinates in a similar way. As an illustration, we give an explicit example to show that the Cayley projective plane is not Desarguean.     

Chapter 8: invariant preserving linear matrix mappings over rings and related topics

This paper surveys the ideas involved in the theory of invariant preserving linear mappings of matrixrings where the scalar ring is not necessarily a field. Section 1 provides several historical examples of the origins of these problems. Section 2 discusses the basic context when the vector space over a field is replaced by a projective module over a commutative ring. Section 3 sketches the classification of the rank one preserving linear mappings using the approach of McDonald, Marcus, and Moyls. Section 4 continues the discussion of Section 3 by placing the problem within the context of group schemes and the invariant preserving theory of Waierhousc. Section 5 begins a sketch of the evolution of these ideas to a context where the scalar ring h not necessarily commutative with a discussion of some classical results of Hua concerning coherence, projective geometry, and matrices over division rings. Trie concluding section, Section 6, discusses the results developed by Wong of linear preserving maps over noncommutative scaiar rings.     

Quotients of incidence geometries

We develop a theory for quotients of geometries and obtain sufficient conditions for the quotient of a geometry to be a geometry. These conditions are compared with earlier work on quotients, in particular by Pasini and Tits. We also explore geometric properties such as connectivity, firmness and transitivity conditions to determine when they are preserved under the quotienting operation. We show that the class of coset pregeometries, which contains all flag-transitive geometries, is closed under an appropriate quotienting operation.     

Additive symmetric bijections on geometric structures

Motivated by their existence in the algebraic theory of quadratic forms we define and study additive symmetric bijections on geometric structures. We give a construction of these bijections on IP-designs and then show that their existence on a design is equivalent to the design arising as the points and hyperplanes of a projective geometry. Using additive symmetric bijections we then develop a combinatorial framework for the study of finitely generated Witt rings.     

Kinematic formulas for finite lattices

In analogy to valuation characterizations and kinematic formulas of convex geometry, we develop a combinatorial theory of invariant valuations and kinematic formulas for finite lattices. Combinatorial kinematic formulas are shown to have application to some probabilistic questions, leading in turn to polynomial identities for Möbius functions and Whitney numbers.     

Logical and graph-theoretic characterizations of the self-dual sentences of projective geometry

The self-dual sentences of projective geometry are characterized in terms of their logical structure. This syntactical analysis reveals an unexpected relationship with graph theory, the self-dual sentences being precisely the geometric interpretations (in a reasonably natural sense) of arbitrary sentences of graph theory.     

On a class of projectively Ricci-flat Finsler metrics

Every Finsler metric induces a spray on a manifold. With a volume form on a manifold, every spray can be deformed to a projective spray. The Ricci curvature of a projective spray is called the projective Ricci curvature. The projective Ricci curvature is an important projective invariant in Finsler geometry. In this paper, we study and characterize projectively Ricci-flat square metrics. Moreover, we construct some nontrivial examples on such Finsler metrics.