Sphere Packings, Lattices and Groups

Distance-Regular Graphs

Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

Preface 1. Introduction 2. Graph operations and modifications 3. Spectrum and structure 4. Characterizations by spectra 5. Structure and one eigenvalue 6. Spectral techniques 7. Laplacians 8. Additional topics 9. Applications Appendix Bibliography Index of symbols Index.

An Introduction to the Theory of Graph Spectra: Spectral techniques

An introduction to noncommutative noetherian rings

Dense near polygons

We introduce the notion of valuation of a dense near polygon. The valuations of a dense near polygon $F$ describe the possible relations between a point of a dense near polygon $\cS$ and any geodetically closed sub near polygon of $\cS$ isomorphic to $F$. Several nice properties of valuations are given and several classes of these objects are defined. Valuations are an important tool for classifying dense near polygons.

Valuations of near polygons

We present a common construction for some known infinite classes of generalized quadrangles. Whether this construction yields other (unknown) generalized quadrangles is an open problem. The class of generalized quadrangles obtained this way is characterized in two different ways. On the one hand, they are exactly the generalized quadrangles having a spread of symmetry. On the other hand, they can be characterized in terms of the group of projectivities with respect to a spread. We explore some properties of these generalized quadrangles. All these results can be applied to the theory of the glued near hexagons, a class of near hexagons introduced by the author in De Bruyn (1998) On near hexagons and spreads of generalized quadrangles, preprint.

Generalized Quadrangles with a Spread of Symmetry

Let ? be a thick dual polar space of rank n ? 2 admitting a full polarized embedding e in a finite-dimensional projective space ?, i.e., for every point x of ?, e maps the set of points of ? at non-maximal distance from x into a hyperplane e?(x) of ?. Using a result of Kasikova and Shult [11], we are able the show that there exists up to isomorphisms a unique full polarized embedding of ? of minimal dimension. We also show that e? realizes a full polarized embedding of ? into a subspace of the dual of ?, and that e? is isomorphic to the minimal full polarized embedding of ?. In the final section, we will determine the minimal full polarized embeddings of the finite dual polar spaces DQ(2n,q), DQ ?(2n+1,q), DH(2n?1,q 2) and DW(2n?1,q) (q odd), but the latter only for n? 5. We shall prove that the minimal full polarized embeddings of DQ(2n,q), DQ ?(2n+1,q) and DH(2n?1,q 2) are the `natural' ones, whereas this is not always the case for DW(2n?1, q).

/pdf/minimal-full-polarized-embeddings-of-dual-polar-spaces-184slieo0n.pdf

Bart De Bruyn

Papers

Dense near polygons

Valuations of near polygons

Generalized Quadrangles with a Spread of Symmetry

Minimal full polarized embeddings of dual polar spaces

Valuations and hyperplanes of dual polar spaces