A normal form and schemes of quadratic forms

We present a solution of the problem of construction of a normal diagonal form for quadratic forms over a local principal ideal ring R = 2R with a QF-scheme of order 2. We give a combinatorial representation for the number of classes of projective congruence quadrics of the projective space over R with nilpotent maximal ideal. For the projective planes, the enumeration of quadrics up to projective equivalence is given; we also consider the projective planes over rings with nonprincipal maximal ideal. We consider the normal form of quadratic forms over the field of p-adic numbers. The corresponding QF-schemes have order 4 or 8. Some open problems for QF-schemes are mentioned. The distinguished finite QF-schemes of local and elementary types (of arbitrarily large order) are realized as the QF-schemes of a field. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 1, pp. 161–178, 2007.     

Bilinear separation of two sets inn-space

The NP-complete problem of determining whether two disjoint point sets in then-dimensional real spaceR ⁿ can be separated by two planes is cast as a bilinear program, that is minimizing the scalar product of two linear functions on a polyhedral set. The bilinear program, which has a vertex solution, is processed by an iterative linear programming algorithm that terminates in a finite number of steps a point satisfying a necessary optimality condition or at a global minimum. Encouraging computational experience on a number of test problems is reported.This material is based on research supported by Air Force Office of Scientific Research grant AFOSR-89-0410, National Science Foundation grant CCR-9101801, and Air Force Laboratory Graduate Fellowship SSN 531-56-2969.     

First-order inertial algorithms involving dry friction damping

We present a Branch-and-Cut algorithm for a class of nonlinear chance-constrained mathematical optimization problems with a finite number of scenarios. Unsatisfied scenarios can enter a recovery mode. This class corresponds to problems that can be reformulated as deterministic convex mixed-integer nonlinear programming problems with indicator variables and continuous scenario variables, but the size of the reformulation is large and quickly becomes impractical as the number of scenarios grows. The Branch-and-Cut algorithm is based on an implicit Benders decomposition scheme, where we generate cutting planes as outer approximation cuts from the projection of the feasible region on suitable subspaces. The size of the master problem in our scheme is much smaller than the deterministic reformulation of the chance-constrained problem. We apply the Branch-and-Cut algorithm to the mid-term hydro scheduling problem, for which we propose a chance-constrained formulation. A computational study using data from ten hydroplants in Greece shows that the proposed methodology solves instances faster than applying a general-purpose solver for convex mixed-integer nonlinear programming problems to the deterministic reformulation, and scales much better with the number of scenarios.

     

Extended dual affine planes

We study a class of diagram geometries, achieve a characterization of extended dual affine planes, and embed extended dual affine planes in extended projective planes. The geometries studied are rank 3 diagram geometries such that the residue of a point is a dual net, and the residue of a plane is linear; the dual of such a geometry has partitions on lines and planes which are reminiscent of parallelism of lines and planes of an affine 3-space. Examples of these geometries (some in dual form) include extended dual affine planes, Laguerre planes, 3-nets, and orthogonal arrays of strength 3. Theorem: Any such finite geometry satisfying Buekenhout's intersection property, and such that any two points are coplanar, is an extended dual affine plane (and has order 2, 4, or 10). Theorem: This geometry may be embedded in an extended projective plane of the same order.This research was partially supported by NSF Grant MCS-8102361.     

Some remarks on orders of projective planes,planar difference sets and multipliers

In this paper we investigate how finite group theory, number theory, together with the geometry of substructures can be used in the study of finite projective planes. Some remarks concerning the function v(x)= x ² + x + 1are presented, for example, how the geometry of a subplane affects the factorization of v(x). The rest of this paper studies abelian planar difference sets by multipliers.Partially supported by NSA grant MDA904-90-H-1013.     

Polarities in n-uniform projective Hjelmslev planes

In [9] the author has studied polarities in finite 2-uniform projective Hjelmslev planes. The present paper deals with polarities in finite n-uniform projective Hjelmslev planes (n 2).The author's research was supported by IWONL grant no. 840037.     

Complexity of the Isomorphism Problem for Computable Free Projective Planes of Finite Rank

Studying computable representations of projective planes, we prove that the isomorphism problem in the class of free projective planes of finite rank is an m-complete Δ⁰₃-set within the class.     

On the embedding of some linear spaces in finite projective planes

The problem of embedding of linear spaces in finite projective planes has been examined by several authors ([1], [2], [3], [4], [5], [6]). In particular, it has been proved in [1] that a linear space which is the complement of a projective or affine subplane of order m is embeddable in a unique way in a projective plane of order n. In this article, we give a generalization of this result by embedding linear spaces in a finite projective plane of order n, which are complements of certain regularA-affine linear spaces with respect to a finite projective plane.     

Barbilian planes

An axiomatization of a class of planes generalizing projective planes is given. With a Moufang condition, these planes have coordinates from an arbitrary alternative ring in whichab=1 impliesba=1. The new techniques include covering planes, homotopy, tangent bundles, and the Lie ring of sections of the tangent bundle, although there is no topological, differential, or algebraic geometric structure.This research was supported in part by NSF grant DMS 850697-01.     

A nearfield-free definition of regular Hughes planes and some embeddings of PG(3,q) in Hughes planes of order q 4

We give a nearfield-free definition of some finite and infinite incidence systems by means of half-points and half-lines and show that they are projective planes. We determine a planar ternary ring for these planes and use it to determine the full collineation group and to demonstrate some embeddings of these planes among themselves. We show that these planes include all finite regular Hughes planes and many infinite ones. We also show that PG(3, q) embeds in Hu(q ⁴) (and show infinite versions of this embedding). Dan Hughes 80th Birthday.     

Degree 8 Maximal Arcs in PG(2,2), h Odd

In a recent paper R. Mathon gave a new construction method for maximal arcs in finite Desarguesian projective planes that generalised a construction of Denniston. He also gave several instances of the method to construct new maximal arcs. In this paper, the structure of the maximal arcs is examined to give geometric and algebraic methods for proving when the maximal arcs are not of Denniston type. New degree 8 maximal arcs are also constructed in PG(2,2^h), h5, h odd. This, combined with previous results, shows that every Desarguesian projective plane of (even) order greater that 8 contains a degree 8 maximal arc that is not of Denniston type.     

über Kurven vom TYP (m;n) und Beispiele total m-regulärer (k,n)-Kurven

In this paper we are concerned with a special kind of subsets of finite projective planes and give some new examples of totally m-regular (k,n)-arcs.

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Herrn Professor Dr. WERNER BURAU zum 70. Geburtstag     

An accelerated covering relaxation algorithm for solving 0–1 positive polynomial programs

The purpose of this note is to present an accelerated algorithm for solving 0–1 positive polynomial (PP) problems. Like our covering relaxation algorithm (Management Science 1979), the accelerated algorithm is a cutting plane method, which uses the linear set covering problem as a relaxation for PP. However, a unique and novel feature of the accelerated algorithm is that it attempts to generate cutting planes from heuristic solutions to the set covering problem whenever possible. Computational results reveal that this strategy of generating cutting planes has led to a significant reduction in the computational time required to solve a PP problem.This research was partially supported by National Sciences and Engineering Research Council Canada Grants 67-4181 and 67-3998, Office of Naval Research Contract N00014-76-C-0418, and National Science Foundation Grant ECS80-22027.     

On the maximum number of different ordered pairs of symbols in sets of latin squares

In this paper, we study the problem of constructing sets of s latin squares of order m such that the average number of different ordered pairs obtained by superimposing two of the s squares in the set is as large as possible. We solve this problem (for all s) when m is a prime power by using projective planes. We also present upper and lower bounds for all positive integers m. © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 1–15, 2005.     

On tactical decompositions of class number 2

In this article we consider tactical decompositions of class number 2 of symmetric designs. Our main result says that if the orders are prime, then the only decompositions are of affine type. Moreover, we study symmetric decompositions of finite projective planes and show that, except in some cases, they are related to Baer subplanes, unitals, or 2 - ((m ² - m + 1)m, m, 1)designs.     

Finite Homogeneous 3-Graphs

By “3-graph” we mean a pair (V, E) such that E ? [V]³. We show that the only non-trivial finite 3-graphs homogeneous in the sense of Fraïssé are those associated with the projective planes over GF(2) and GF(3), and with the projective lines over GF(5) and GF(9). To exclude other possibilities we use the classification of doubly transitive finite permutation groups.     

Linear Perfect Codes and a Characterization of the Classical Designs

A new definition for the dimension of a combinatorial t-(v,k,) design over a finite field is proposed. The complementary designs of the hyperplanes in a finite projective or affine geometry, and the finite Desarguesian planes in particular, are characterized as the unique (up to isomorphism) designs with the given parameters and minimum dimension. This generalizes a well-known characterization of the binary hyperplane designs in terms of their minimum 2-rank. The proof utilizes the q-ary analogue of the Hamming code, and a group-theoretic characterization of the classical designs.     

Linear representation of a class of projective planes in a four dimensional projective space

Summary The theory of linear representations of projective planes developed by Bruck and one of the authors (Bose) in two earlier papers [J. Algebra1 (1964), pp. 85–102 and4 (1966), pp. 117–172] can be further extended by generalizing the concept of incidence adopted there. A linear representation is obtained for a class of non-Desarguesian projective planes illustrating this concept of generalized incidence. It is shown that in the finite case, the planes represented by the new construction are derived planes in the sense defined by Ostrom [Trans. Amer. Math Soc.111 (1964), pp. 1–18] and Albert [Boletin Soc. Mat. Mex,11 (1966), pp, 1–13] of the dual of translation planes which can be represented in a 4-space by the Bose-Bruck construction. An analogous interpretation is possible for the infinite case. This research was sponsored by the National Science Foundation under Grant No. GP-8624, and the U.S. Air Force Office of Scientific Research under Grant No. AFOSR-68-1406. This research was conducted while the author was visiting professor at the University of North Carolina at Chapel Hill. His research was also partially supported by C.N.R. Entrata in Redazione il 28 maggio 1970.     

Affine planes over finite rings,a summary

In Keppens (Innov. Incidence Geom. 15: 119–139, 2017) we gave a state of the art concerning “projective planes” over finite rings. The current paper gives a complementary overview for “affine planes” over rings (including the important subclass of desarguesian affine Klingenberg and Hjelmslev planes). No essentially new material is presented here but we give a summary of known results with special attention to the finite case, filling a gap in the literature.     

Branch-and-price-and-cut on the clique partitioning problem with minimum clique size requirement

Given a complete graph K_n=(V,E)with edge weight c_e on each edge, we consider the problem of partitioning the vertices of graph K_n into subcliques that have at least S vertices, so as to minimize the total weight of the edges that have both endpoints in the same subclique. In this paper, we consider using the branch-and-price method to solve the problem. We demonstrate the necessity of cutting planes for this problem and suggest effective ways of adding cutting planes in the branch-and-price framework. The NP hard pricing problem is solved as an integer programming problem. We present computational results on large randomly generated problems.