Configuration Theorem Research Articles

THE GROUP CONFIGURATION THEOREM FOR GENERICALLY STABLE TYPES

Abstract We generalize Hrushovski’s group configuration theorem to the case where the type of the configuration is generically stable, without assuming tameness of the ambient theory. The properties of generically stable types, which we recall in the second section, enable us to adapt the proof known in the stable context.

Projective geometries arising from Elekes-Szabó problems

We generalise the Elekes-Szab\'o theorem to arbitrary arity and dimension and characterise the complex algebraic varieties without power saving. The characterisation involves certain algebraic subgroups of commutative algebraic groups endowed with an extra structure arising from a skew field of endomorphisms. We also extend the Erd\H{o}s-Szemer\'edi sum-product phenomenon to elliptic curves. Our approach is based on Hrushovski's framework of pseudo-finite dimensions and the abelian group configuration theorem.

Four Lines and a Rectangle

This paper presents some configuration theorems concerning rectangles inscribed in four lines.

Dimensional lower bounds for Falconer type incidence theorems

Let $1 \leq k \leq d$ and consider a subset $E\subset \mathbb{R}^d$. In this paper, we study the problem of how large the Hausdorff dimension of $E$ must be in order for the set of distinct noncongruent $k$-simplices in $E$ (that is, noncongruent point configurations of $k+1$ points from $E$) to have positive Lebesgue measure. This generalizes the $k=1$ case, the well-known Falconer distance problem and a major open problem in geometric measure theory. We establish a dimensional lower threshold of $\frac{d(k+1)}{d+2}$ for Falconer type theorems for $k$-simplices. This threshold is nontrivial in the range $d/2 \leq k \leq d$ and is obtained through counting simplices in a standard lattice using results of the Gauss circle problem. Many results on Falconer type theorems have been established through incidence theorems, which generally establish sufficient but not necessary conditions for the point configuration theorems. We also establish a dimensional lower threshold of $\frac{d+1}{2}$ on incidence theorems for $k$-simplices where $k\leq d \leq 2k+1$ by generalizing an example of Mattila. Finally, we prove a dimensional lower threshold of $\frac{d+1}{2}$ on incidence theorems for triangles in a convex setting in every dimension greater than $3$. This last result generalizes work by Iosevich and Senger on distances that was built on a construction by Valtr. The final result utilizes number-theoretic machinery to estimate the number of solutions to a Diophantine equation.

On a class of partial planes related to biplanes

In this note we consider partial planes in which for each element x (point or line) there exists a unique opposite element or antipode x* which cannot be joined to x or has no intersection with x. We also require the existence of a triangle. Such partial planes will be called antipodal planes. We are mainly interested in the subclass of regular antipodal planes satisfying: p I L implies p* I L* for all points p and lines L. We shall provide a free construction of infinite regular antipodal planes. The objects thus constructed are not free objects in the usual sense since between antipodal planes there do not exist proper homomorphisms. On the other hand, regular antipodal planes do have a canonical homomorphic image which is a biplane (cf. Payne, J Comb Theory A 12:268–282, 1972). Regular antipodal planes can be coordinatized by certain algebraic systems in a similar way as projective planes are coordinatized by ternary rings. Again by a free construction, we shall provide examples satisfying a configuration theorem comparable to the Fano condition with fixed line at infinity.

Quasiminimal structures, groups and Zariski-like geometries

We generalize Hrushovski's Group Configuration Theorem to quasiminimal classes. As an application, we present Zariski-like structures, a generalization of Zariski geometries, and show that a group can be found there if the pregeometry obtained from the bounded closure operator is non-trivial.

Skewers

The skewer of a pair of skew lines in space is their common perpendicular. To configuration theorems of plane projective geometry involving points and lines (such as Pappus or Desargues) there correspond configuration theorems in space: points and lines in the plane are replaced by lines is space, the incidence between a line and a point translates as the intersection of two lines at right angle, and the operations of connecting two points by a line or by intersecting two lines at a point translate as taking the skewer of two lines. These configuration theorems hold in elliptic, Euclidean, and hyperbolic geometries. This correspondence principle extends to plane configuration theorems involving polarity. For example, the theorem that the three altitudes of a triangle are concurrent corresponds to the Petersen–Morley theorem that the common normals of the opposite sides of a space right-angled hexagon have a common normal. We define analogs of plane circles (they are 2-parameter families of lines in space) and extend the correspondence principle to plane theorems involving circles. We also discuss the skewer versions of the Sylvester problem: given a finite collection of pairwise skew lines such that the skewer of any pair intersects at least one other line at right angle, do all lines have to share a skewer? The answer is positive in the elliptic and Euclidean geometries, but negative in the hyperbolic one.

Study on Deformation Inspection System of Car Seat Frame Based on Virtual Instrument

An automatic deformation inspection system of car seat frame based on virtual instrument is invented and applied in real production. The hardware configuration, software modules, inspection and control theorem and application example are introduced. The system is easy to use with sound real-time performance, stability and high accuracy.

Recovering the hyperdefinable group action in the group configuration theorem

AbstractIn this paper, we continue the construction done in [3], so that under model-4-CA or 4-CA, given a bounded quadrangle C induced from a group configuration, we build a canonical hyperdefinable homogeneous space equivalent to C. When C is principal, we can choose the homogeneous space principal as well.

Generalizations of small profinite structures

AbstractWe generalize the model theory of small profinite structures developed by Newelski to the case of compact metric spaces considered together with compact groups of homeomorphisms and satisfying the existence of m-independent extensions (we call them compact e-structures). We analyze the relationships between smallness and different versions of the assumption of the existence of m-independent extensions and we obtain some topological consequences of these assumptions. Using them, we adopt Newelski's proofs of various results about small profinite structures to compact e-structures. In particular, we notice that a variant of the group configuration theorem holds in this context.A general construction of compact structures is described. Using it, a class of examples of compact e-structures which are not small is constructed.It is also noticed that in an m-stable compact e-structure every orbit is equidominant with a product of m-regular orbits.

A remark on multivalued algebraic groups

We point out how suitable algebraic n-valued groups (in the sense of Buchstaber) give rise, in a reasonably canonical manner, to algebraic groups. This is proved using the group configuration theorem of Hrushovski. In particular this applies to all algebraic 2-valued groups.

CONSTRUCTING AN ALMOST HYPERDEFINABLE GROUP

This paper completes the proof of the group configuration theorem for simple theories started in [1]. We introduce the notion of an almost hyperdefinable (poly-)structure, and show that it has a reasonable model theory. We then construct an almost hyperdefinable group from a polygroup chunk.

APPLICATIONS OF THE GROUP CONFIGURATION THEOREM IN SIMPLE THEORIES

We reconstruct the group action in the group configuration theorem. We apply it to show that in an ω-categorical theory a finitely based pseudolinear regular type is locally modular, and the geometry associated to a finitely based locally modular regular type is projective geometry over a finite field.

Packing paths in digraphs

AbstractLet ${\cal G}$ be a fixed set of digraphs. Given a digraph H, a ${\cal G}$‐packing in H is a collection ${\cal P}$ of vertex disjoint subgraphs of H, each isomorphic to a member of ${\cal G}$. A ${\cal G}$‐packing ${\cal P}$ is maximum if the number of vertices belonging to members of ${\cal P}$ is maximum, over all ${\cal G}$‐packings. The analogous problem for undirected graphs has been extensively studied in the literature. The purpose of this paper is to initiate the study of digraph packing problems. We focus on the case when ${\cal G}$ is a family of directed paths. We show that unless ${\cal G}$ is (essentially) either $\{ \vec {P}_1 \}$, or $\{ \vec {P}_1, \vec {P}_2 \}$, the G‐packing problem is NP‐complete. When ${\cal G} = \{ \vec {P}_1 \}$, the ${\cal G}$‐packing problem is simply the matching problem. We treat in detail the one remaining case, ${\cal G} = \{ \vec {P}_1, \vec {P}_2 \}$. We give in this case a polynomial algorithm for the packing problem. We also give the following positive results: a Berge type augmenting configuration theorem, a min‐max characterization, and a reduction to bipartite matching. These results apply also to packings by the family ${\cal G}$ consisting of all directed paths and cycles. We also explore weighted variants of the problem and include a polyhedral analysis. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 81–94, 2003

Group configurations and germs in simple theories

AbstractWe develop the theory of germs of generic functions in simple theories. Starting with an algebraic quadrangle (or other similar hypotheses), we obtain an “almost” generic group chunk, where the product is defined up to a bounded number of possible values. This is the first step towards the proof of the group configuration theorem for simple theories, which is completed in [3].

The Group Configuration in Simple Theories and its Applications

AbstractIn recent work, the authors have established the group configuration theorem for simple theories, as well as some of its main applications from geometric stability theory, such as the binding group theorem, or, in the ω-categorical case, the characterization of the forking geometry of a finitely based non-trivial locally modular regular type as projective geometry over a finite field and the equivalence of pseudolinearity and local modularity.The proof necessitated an extension of the model-theoretic framework to includealmost hyperimaginaries, and the study ofpolygroups.

Dependence relation in pregeometries

The aim of this paper is to set a foundation to separate geometric model theory from model theory. Our goal is to explore the possibility to extend results from geometric model theory to non first order logic (e.g. $L_{\omega_1,\omega}$ ). We introduce a dependence relation between subsets of a pregeometry and show that it satisfies all the formal properties that forking satisfies in simple first order theories. This happens when one is trying to lift forking to nonelementary classes, in contexts where there exist pregeometries but not necessarily a well-behaved dependence relation (see for example [HySh]). We use these to reproduce S. Buechler's characterization of local modularity in general. These results are used by Lessmann to prove an abstract group configuration theorem in [Le2].

On the density of sets of divisors

Consider the lattice of divisors of n, [1, n]. For any downset (ideal) ℐ in [1, n] we get a forbidden configuration theorem of the type that if a set of divisors D avoids certain configurations, then |D|⩾|ℐ|. If we let ℓ be the set of minimal elements of [1, n] not in ℐ, then we forbid in D the configurations C(s) (defined in the paper) for s∈ℒ. This generalizes a result of Alon and in turn generalizes a result of Sauer, Perles and Shelah.

Intersection properties of invariant manifolds in certain twist maps

We consider the spaceN ofC 2 twist maps that satisfy the following requirements. The action is the sum of a purely quadratic term and a periodic potential times a constantk (hereafter called the nonlinearity). The potential restricted to the unit circle is bimodal, i.e. has one local minimum and one local maximum. The following statements are proven for maps inN with nonlinearityk large enough. The intersection of the unstable and stable invariant manifolds to the hyperbolic minimizing periodic points contains minimizing homoclinic points. Consider two finite pieces of these manifolds that connect two adjacent homoclinic minimizing points (hereafter called fundamental domains). We prove that all such fundamental domains have precisely one point in their intersection (the Single Intersection theorem).In addition, we show that limit points of minimizing points are recurrent, which implies that Aubry Mather sets (with irrational rotation number) are contained in diamonds formed by local stable and unstable manifolds of nearby minimizing periodic orbits (the Diamond Configuration theorem). Another corollary concerns the intersection of the minimax orbits with certain symmetry lines of the map.

Forbidden Configurations, Discrepancy and Determinants

An m × n matrix A is said to have hereditary discrepancy d if the maximum over submatrices B of A of the minimum over (−1/2, +1/2)-vectors x of ‖ Bx ‖ ∞ . If A is integral and has no repeated rows then we show that m is bounded by a polynomial in n for fixed d . This improves a result of Lovász and Vesztergombi for fixed d . Let A be an m × n matrix with no repeated rows and with each square submatrix having determinant {0, ±1, ±2,..., ± k } . Then m is bounded by a polynomial in n for fixed k . This extends a result of Heller for totally unimodular matrices. (A result of Kung combined with results of J. Lee provides a much better bound.) Both bounds follow from repeated applications of the forbidden configuration theorem of Sauer, Perles and Shelah, which states that an m × n (0, 1)-matrix with no repeated rows and no 2 k xk submatrix of all (0, 1)-rows on k columns has m at most ( n k − 1 ) + ( n k − 2 ) + ⋯ + ( n 0 ) .