Another Constructive Axiomatization of Euclidean Planes

H. Tietze has proved algebraically that the geometry of uniquely determined ruler and compass constructions coincides with the geometry of ruler and set square constructions. We provide a new proof of this result via new universal axiom systems for Euclidean planes of characteristic ≠ 2 in languages containing only operation symbols.     

Ternary operations as primitive notions for constructive plane geometry III

This paper continues the investigations begun in [6] and continued in [7] about quantifier-free axiomatizations of plane Euclidean geometry using ternary operations. We show that plane Euclidean geometry over Archimedean ordered Euclidean fields can be axiomatized using only two ternary operations if one allows axioms that are not first-order but universal L_w1,w sentences. The operations are: the transport of a segment on a halfline that starts at one of the endpoints of the given segment, and the operation which produces one of the intersection points of a perpendicular on a diameter of a circle (which intersects that diameter at a point inside the circle) with that circle. MSC: 03F65, 51M05, 51M15.     

Constructive Axiomatization of Plane Hyperbolic Geometry

We provide a universal axiom system for plane hyperbolic geometry in a firstorder language with two sorts of individual variables, ‘points’ (upper‐case) and ‘lines’ (lowercase), containing three individual constants, A₀, A₁, A₂, standing for three non‐collinear points, two binary operation symbols, φ and ι, with φ(A, B) = l to be interpreted as ‘𝓁 is the line joining A and B’ (provided that A ≠ B, an arbitrary line, otherwise), and ι(g, h) = P to be interpreted as 𝓁P is the point of intersection of g and h (provided that g and h are distinct and have a point of intersection, an arbitrary point, otherwise), and two binary operation symbols, π₁(P, 𝓁) and —₂(P, 𝓁), with π_i(P, 𝓁) = g (for i = 1, 2) to be interpreted as ‘g is one of the two limiting paralle lines from P to 𝓁 (provided that P is not on 𝓁, an arbitrary line, otherwise).     

A generalization of the Discrete Isoperimetric Inequality for Piecewise Smooth Curves of Constant Geodesic Curvature

Summary The discrete isoperimetric problem is to determine the maximal area polygon with at most <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>k$ vertices and of a given perimeter. It is a classical fact that the unique optimal polygon on the Euclidean plane is the regular one. The same statement for the hyperbolic plane was proved by K\'aroly Bezdek [1] and on the sphere by L\'aszl\'o Fejes T\'oth [3]. In the present paper we extend the discrete isoperimetric inequality for ``polygons' on the three planes of constant curvature bounded by arcs of a given constant geodesic curvature.     

Ternary Operations as Primitive Notions for Constructive Plane Geometry IV

In this paper we provide a quantifier-free constructive axiomatization for Euclidean planes in a first-order language with only ternary operation symbols and three constant symbols (to be interpreted as ‘points’). We also determine the algorithmic theories of some ‘naturally occurring’ plane geometries. Mathematics Subject Classification: 03F65, 51M05, 51M15, 03B30.     

A Classification of 2-Dimensional Laguerre Planes Admitting 3-Dimensional Groups of Automorphisms in the Kernel

We determine, up to isomorphisms, all 2-dimensional Laguerre planes that admit 3-dimensional groups of automorphisms in the kernel of the action on parallel classes.     

Quantifier elimination for elementary geometry and elementary affine geometry

We introduce new first‐order languages for the elementary n‐dimensional geometry and elementary n‐dimensional affine geometry (n ≥ 2), based on extending $\mathsf {FO}(\beta ,\equiv )$ and $\mathsf {FO}(\beta )$, respectively, with new function symbols. Here, β stands for the betweenness relation and ≡ for the congruence relation. We show that the associated theories admit effective quantifier elimination.     

Geometrical Solution to the Fermat Problem with Arbitrary Weights

The prime motivation for the present study is a famous problem, allegedly first formulated in 1643 by Fermat, and the so-called Complementary Problem (CP), proposed but incorrectly solved in 1941 by Courant and Robbins. For a given triangle, Fermat asks for a fourth point such that the sum of its Euclidean distances, each weighted by +1, to the three given points is minimized. CP differs from Fermat in that the weight associated with one of these points is –1 instead of +1. The geometrical approach suggested in 1998 by Krarup for solving CP is here extended to cover any combination of positive and negative weights associated with the vertices of a given triangle. Among the by-products are surprisingly simple correctness proofs of the geometrical constructions of Torricelli (around 1645), Cavalieri (1647), Viviani (1659), Simpson (1750), and Martelli (1998). Furthermore, alternative proofs of Ptolemy's theorem (around A.D. 150) and an observation by Heinen (1834) are provided.     

Strongly non-repetitive sequences and progression-free sets

By a simple method we show the existence of (1) a sequence on two symbols in which no four blocks occur consecutively that are permutations of each other, and (2) a sequence on three symbols in which no three blocks occur consecutively that are permutations of each other. The problem of the existence of a sequence on four symbols in which no two blocks occur consecutively that are permutations of each other remains open.     

Commutators of Bilinear Pseudodifferential Operators and Lipschitz Functions

Commutators of bilinear pseudodifferential operators and the operation of multiplication by a Lipschitz function are studied. The bilinear symbols of the pseudodifferential operators considered belong to classes that are shown to properly contain certain bilinear Hörmander classes of symbols of order one. The corresponding commutators are proved to be bilinear Calderón–Zygmund operators.     

Coding and decoding representations of 3D shapes

The aim of this study is to examine students’ ability in interpreting and constructing plane representations of 3D shapes, and to trace categories of students that reflect different types of behaviour in representing 3D shapes. To achieve this goal, one test was administered to 279 students in grades 5–9, and forty of them were interviewed. The results of the study showed that the representation of 3D shapes is composed of two general representing/cognitive abilities, coding and decoding. Decoding refers to interpreting the structural elements and geometrical properties of 3D shapes in plane representations, while coding refers to constructing plane representations and nets of 3D shapes, and translating from one representational mode to another. A mixed-method analysis showed that four categories of students can be identified that describe four types of behaviour and explain students’ reasoning patterns in representing 3D shapes.     

Topological conjugacy of constant length substitution dynamical systems

Primitive constant length substitutions generate minimal symbolic dynamical systems. In this article we present an algorithm which can produce the list of injective substitutions of the same length that generate topologically conjugate systems. We show that each conjugacy class contains infinitely many substitutions which are not injective. As examples, the Toeplitz conjugacy class contains three injective substitutions (two on two symbols and one on three symbols), and the length two Thue–Morse conjugacy class contains twelve substitutions, among which are two on six symbols. Together, they constitute a list of all primitive substitutions of length two with infinite minimal systems which are factors of the Thue–Morse system.     

Bergman Space Structure,Commutative Algebras of Toeplitz Operators,and Hyperbolic Geometry

We exhibit a surprising but natural connection among the Bergman space structure, commutative algebras of Toeplitz operators and pencils of hyperbolic straight lines. The commutative C*-algebras of Toeplitz operators on the unit disk can be classified as follows. Each pencil of hyperbolic straight lines determines the set of symbols consisting of functions which are constant on corresponding cycles, the orthogonal trajectories to lines forming a pencil. It turns out that the C*-algebra generated by Toeplitz operators with this class of symbols is commutative. Submitted: January 15, 2001?Revised: February 7, 2002     

Castor quadruplorum

The busy beaver problem of Rado [6] is reexamined for the case of Turing machines given by quadruples rather than quintuples. Moreover several printing symbols are allowed. Some values of the corresponding beaver function are given and it is shown that this function for a fixed number of states and varying number of symbols is nonrecursive for three or more states and recursive for two states. As a byproduct we get that the minimal number of states in a universal Turing machine (quadruples) is three.     

Duality in stable planes and related closure and kernel operations

We compare two constructions that dualize a stable plane in some sense, namely the dual plane and the opposite plane. Applying both constructions one after another we obtain a closure or kernel operation, depending on the order of execution.We examine the effect of these constructions on the automorphism group and apply our results in order to compute the automorphism groups of the complex cylinder plane, the complex united cylinder plane, and their duals. Beside the complex projective, affine, and punctured projective plane these planes are in fact the most homogeneous four-dimensional stable planes, as will be shown elsewhere [1].Supported by Studienstiftung des deutschen Volkes.     

Embedding a Latin square with transversal into a projective space

A Latin square of side n defines in a natural way a finite geometry on 3n points, with three lines of size n and n² lines of size 3. A Latin square of side n with a transversal similarly defines a finite geometry on 3n+1 points, with three lines of size n, n²−n lines of size 3, and n concurrent lines of size 4. A collection of k mutually orthogonal Latin squares defines a geometry on kn points, with k lines of size n and n² lines of size k. Extending the work of Bruen and Colbourn [A.A. Bruen, C.J. Colbourn, Transversal designs in classical planes and spaces, J. Combin. Theory Ser. A 92 (2000) 88-94], we characterise embeddings of these finite geometries into projective spaces over skew fields.     

Non-local pseudo-differential operators

The notion of non-local pseudo-differential operators, as well as their symbols and the operation on holomorphic functions, is established and the invertibility theorem for such operators is proved.     

A stabilization law for two semi-infinite interacting strings of characters

We consider a Markov chain that describes the evolution of two interacting strings of symbols. The transitions probalitities of this Markov chain depend only on the rightmost symbols of both strings. The main goal of the present paper is to prove a limit theorem (stabilization law): the distribution of the rightmost symbols converges to some limit correlation function.1 Partially supported by FAPESP (2002/01501-9) and RFBR (02-01-00415)2 Partially supported by RFBR (02-01-00415)     

Binary hyperidentities of lattices

Summary An equational identity of a given type involves two kinds of symbols: individual variables and the operation symbols. For example, the distributive identity: x (y + z) = x y + x z has three variable symbols {x, y, z} and two operation symbols {+, }. Here the variables range over all the elements of the base set while the two operation symbols are fixed. However, we shall say that an identity ishypersatisfied by a varietyV if, whenever we also allow the operation symbols to range over all polynomials of appropriate arity, the resulting identities are all satisfied byV in the usual sense. For example, the ring of integers Z; +, satisfies the above distributive law, but it does not hypersatisfy the same formal law because, e.g., the identityx + (y z) = (x + y) (x + z) is not valid. By contrast, is hypersatisfied by the variety of all distributive lattices and is thus referred to as a distributive latticehyperidentity. Thus a hyperidentity may be viewed as an equational scheme for writing a class of identities of a given type and the original identities themselves are obtained as special cases by substituting specific polynomials of appropriate arity for the operation symbols in the scheme. In this paper, we provide afinite equational scheme which is a basis for the set of all binary lattice hyperidentities of type 2, 2, .This research was supported by the NSERC operating grant # 8215     

On the Limit Set of Discrete Subgroups of PU(2,1)

Let G be a discrete subgroup of PU(2,1); G acts on preserving the unit ball , equipped with the Bergman metric. Let be the limit set of G in the sense of Chen–Greenberg, and let be the limit set of the G-action on in the sense of Kulkarni. We prove that L(G) = Λ(G) ∩ S ³ and Λ(G) is the union of all complex projective lines in which are tangent to S ³ at a point in L(G).