By Marton Elekes, Miklos Laczkovich

Enable ℝℝ denote the set of actual valued features outlined at the genuine line. A map D: ℝℝ → ℝℝ is related to be a distinction operator if there are actual numbers a i, b i (i = 1, :, n) such that (Dƒ)(x) = ∑ i=1 n a i ƒ(x + b i) for each ƒ ∈ ℝℝand x ∈ ℝ. by means of a process of distinction equations we suggest a collection of equations S = {D i ƒ = g i: i ∈ I}, the place I is an arbitrary set of indices, D i is a distinction operator and g i is a given functionality for each i ∈ I, and ƒ is the unknown functionality. possible turn out approach S is solvable if and provided that each finite subsystem of S is solvable. besides the fact that, if we glance for recommendations belonging to a given type of features then the analogous assertion isn't any longer actual. for instance, there exists a method S such that each finite subsystem of S has an answer that's a trigonometric polynomial, yet S has no such resolution; additionally, S has no measurable recommendations. This phenomenon motivates the next definition. enable be a category of services. The solvability cardinal sc( ) of is the smallest cardinal quantity κ such that every time S is a procedure of distinction equations and every subsystem of S of cardinality lower than κ has an answer in , then S itself has an answer in . during this paper we make certain the solvability cardinals of so much functionality periods that take place in research. because it seems, the behaviour of sc( ) is quite erratic. for instance, sc(polynomials) = three yet sc(trigonometric polynomials) = ω 1, sc({ƒ: ƒ is continuous}) = ω 1 yet sc({f : f is Darboux}) = (2 ω )+, and sc(ℝℝ) = ω. We constantly be sure the solvability cardinals of the sessions of Borel, Lebesgue and Baire measurable features, and provides a few partial solutions for the Baire classification 1 and Baire type α capabilities.

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A point not contained in S is a geometrical interior point of 5 if every line containing it is a secant of S. A point not in S is a geometrical exterior point if it is contained in an exterior line. An arc in a linear space is a nonempty set of points such that no three points in the set are collinear. An arc is called maximal if it is not contained in a larger arc. An arc is called an oval if every one of its points is contained in exactly one tangent line. A Jordan curve in an ideal flat linear space is called a convex curve if every line intersects the curve in no, 1, or two points, or in an interval.

The line at infinity of this Euclidean plane is the unit circle. A parallel class of lines corresponds to the set of Euclidean line and circle segments through two antipodal points on the unit circle. 1 shows an embedding of Desargues' configuration in this 'disk' model of PG(2, R). Fig. 1. The disk model In general, a disk model of a flat projective plane is a geometry on a closed topological disk D together with a fixed-point-free involutory homeomorphism 7 of the boundary of D to itself. Its points are the interior points of the disk together with the set of pairs of boundary points that get exchanged by 7.

1 R 2 -Planes We define an R 2 -plane to be a linear space on a surface homeomorphic to R 2 . Any R- or S-line in R 2 separates R 2 into two connected components. Clearly, given an R- or S-line L and an S-line K in R 2 such that L contains points in the two different connected components of R 2 \ K, L and K have at least two points in common. Since two lines in a linear space intersect in at most one point, we see that an R2-plane cannot contain S-lines. Fig. 5. More than one parallel line The Euclidean plane is the classical example of an R2-plane.

### A cardinal number connected to the solvability of systems of difference equations in a given function class by Marton Elekes, Miklos Laczkovich

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