On the equivalence of Lagrange’s axiom to the Lotschnittaxiom

We prove that, in Hilbert’s plane absolute geometry, an axiom used by Lagrange in a proof of the Euclidean parallel postulate in a paper read on 3 February 1806 at the Institut de France, which states that “If a and b are two parallels from P to g, then the reflection of a in b is parallel to g as well”, is equivalent to F. Bachmann’s Lotschnittaxiom, which states that “The perpendiculars on the sides a right angle always intersect.”     

Combinatorial analysis of proofs in projective and affine geometry

The axioms of projective and affine plane geometry are turned into rules of proof by which formal derivations are constructed. The rules act only on atomic formulas. It is shown that proof search for the derivability of atomic cases from atomic assumptions by these rules terminates (i.e., solves the word problem). This decision method is based on the central result of the combinatorial analysis of derivations by the geometric rules: The geometric objects that occur in derivations by the rules can be restricted to those known from the assumptions and cases. This “subterm property” is proved by permuting suitably the order of application of the geometric rules. As an example of the decision method, it is shown that there cannot exist a derivation of Euclid’s fifth postulate if the rule that corresponds to the uniqueness of the parallel line construction is taken away from the system of plane affine geometry.     

Real numbers and projective spaces: Intuitionistic reasoning with undecidable basic relations

Brouwer introduced in 1924 the notion of an apartness relation for real numbers, with the idea that whenever it holds, a finite computation verifies it in contrast to equality. The idea was followed in Heyting’s axiomatization of intuitionistic projective geometry. Brouwer in turn worked out an intuitionistic theory of “virtual order.” It is shown that Brouwer’s proof of the equivalence of virtual and maximal order goes only in one direction, and that Heyting’s axiomatization needs to be made a bit stronger.     

A geometric proof of a theorem on antiregularity of generalized quadrangles

A geometric proof is given in terms of Laguerre geometry of the theorem of Bagchi, Brouwer and Wilbrink, which states that if a generalized quadrangle of order s > 1 has an antiregular point then all of its points are antiregular.     

Markov random fields,Markov cocycles and the 3-colored chessboard

The well-known Hammersley–Clifford Theorem states (under certain conditions) that any Markov random field is a Gibbs state for a nearest neighbor interaction. In this paper we study Markov random fields for which the proof of the Hammersley–Clifford Theorem does not apply. Following Petersen and Schmidt we utilize the formalism of cocycles for the homoclinic equivalence relation and introduce “Markov cocycles”, reparametrizations of Markov specifications. The main part of this paper exploits this to deduce the conclusion of the Hammersley–Clifford Theorem for a family of Markov random fields which are outside the theorem’s purview where the underlying graph is Z_d. This family includes all Markov random fields whose support is the d-dimensional “3-colored chessboard”. On the other extreme, we construct a family of shift-invariant Markov random fields which are not given by any finite range shift-invariant interaction.     

Kronecker’s density theorem and irrational numbers in constructive reverse mathematics

To prove Kronecker’s density theorem in Bishop-style constructive analysis one needs to define an irrational number as a real number that is bounded away from each rational number. In fact, once one understands “irrational” merely as “not rational”, then the theorem becomes equivalent to Markov’s principle. To see this we undertake a systematic classification, in the vein of constructive reverse mathematics, of logical combinations of “rational” and “irrational” as predicates of real numbers.     

On the Sperner lemma and some colorings of graphs

In this journal, Daniel I. A. Cohen [2] gave a proof of the strong Sperner lemma based on “search” techniques of paths. Klaus Wagner [5] associates a particular tree to a labeled subdivision of a geometric simplex and proves that lemma by counting the number of vertices with an odd degree of that tree. In this paper we prove a general theorem about colorings of simplicial complexes by using a method similar to Wagner's. This theorem implies the strong Sperner lemma and can be used to study some colorings of graphs.     

Differentiability of mappings in the geometry of Carnot manifolds

We study the differentiability of mappings in the geometry of Carnot-Carathéodory spaces under the condition of minimal smoothness of vector fields. We introduce a new concept of hc-differentiability and prove the hc-differentiability of Lipschitz mappings of Carnot-Carathéodory spaces (a generalization of Rademacher’s theorem) and a generalization of Stepanov’s theorem. As a consequence, we obtain the hc-differentiability almost everywhere of the quasiconformal mappings of Carnot-Carathéodory spaces. We establish the hc-differentiability of rectifiable curves by way of proof. Moreover, the paper contains a new proof of the functorial property of the correspondence “a local basis ? the nilpotent tangent cone.”     

Theorems of Barth–Lefschetz type in Sasakian geometry

In this paper, we obtain theorems of Barth–Lefschetz type in Sasakian geometry. As a corollary, this gives a new proof of a classical theorem due to J. Milnor. It also implies connectedness principle and Frankel's type theorem.     

A note on Alberti’s Luzin-type theorem for gradients

Ricerche di Matematica - We give a “soft” proof of Alberti’s Luzin-type theorem in Alberti (J Funct Anal 100:110–118, 1991), using elementary geometric measure theory and...     

A few applications of negative- type inequalities

Applying the negative-type inequalities for square Euclidean distance, we present (1) a parallelatope theorem (a generalization of the parallelogram theorem), (2) a short proof of Rankin’s theorem for the maximum number of dispersed points on a sphere, and (3) a proof of impossibility of a certain geometric embedding for some graphs.     

Moving-centre monotonicity formulae for minimal submanifolds and related equations

Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball, then its area is at least that of the equatorial disk. Recently Brendle and Hung proved a sharp area bound for minimal submanifolds when the prescribed point is not the centre of the ball, which resolved a conjecture of Alexander, Hoffman and Osserman. Their proof involves asymptotic analysis of an ingeniously chosen vector field, and the divergence theorem.In this article we prove a sharp ‘moving-centre’ monotonicity formula for minimal submanifolds, which implies the aforementioned area bound. We also describe similar moving-centre monotonicity formulae for stationary p-harmonic maps, mean curvature flow and the harmonic map heat flow.     

Reclassifying the antithesis of Specker’s theorem

It is shown that a principle, which can be seen as a constructivised version of sequential compactness, is equivalent to a form of Brouwer’s fan theorem. The complexity of the latter depends on the geometry of the spaces involved in the former.     

Comportement asymptotique des fonctions harmoniques en courbure négative

LetM be a complete simply connected Riemannian manifold whose sectional curvatures are bounded between two negative constants. It is shown that, for a given harmonic function onM, non-tangential properties of convergence, boundedness and finiteness of energy are equivalent for almost every point of the geometric boundary. This is a “geometric” analogue of Calderón-Stein theorem in the euclidean half-space. The proof is using Brownian motion, like J. Brossard's one for the euclidean case.     

Théorème de Gabrielov et fonctions log-exp-algébriques

We give a new proof of Wilkie 's theorem on log-exp algebraic functions. The tools used here are “explicit” Gabrielov's theorem and our geometric presentation of the theorem of van den Dries, Macintyre, and Marker on log-exp analytic functions.     

An n-dimensional version of Steinhaus' chessboard theorem

The main result of this paper is an n-dimensional version of the Steinhaus' chessboard theorem. Our theorem implies the Poincaré theorem as well as its parametric extension. But it is known that the Poincaré theorem is equivalent to the Brouwer Fixed-Point theorem.     

Homeomorphisms of the Heisenberg group preserving BMO

We provide a new geometric proof of Reimann’s theorem characterizing quasiconformal mappings as the ones preserving functions of bounded mean oscillation. While our proof is new already in the Euclidean spaces, it is applicable in Heisenberg groups as well as in more general stratified nilpotent Carnot groups.     

The principle of duality in Euclidean and in absolute geometry

In Euclidean geometry and in absolute geometry fragments of the principle of duality hold. Bachmann (Aufbau der Geometrie aus dem Spiegelungsbegriff, 1973, §3.9) posed the problem to find a general theorem which describes the extent of an allowed dualization. It is the aim of this paper to solve this problem. To this end a first-order axiomatization of Euclidean (resp. absolute) geometry is provided which allows the application of Gödel’s Completeness Theorem for first-order logic and the solution of Bachmann’s problem.     

Mutation on knots and Whitney’s 2-isomorphism theorem

Whitney’s 2-switching theorem states that any two embeddings of a 2-connected planar graph in S ² can be connected via a sequence of simple operations, named 2-switching. In this paper, we obtain two operations on planar graphs from the view point of knot theory, which we will term “twisting” and “2-switching” respectively. With the twisting operation, we give a pure geometrical proof of Whitney’s 2-switching theorem. As an application, we obtain some relationships between two knots which correspond to the same signed planar graph. Besides, we also give a necessary and sufficient condition to test whether a pair of reduced alternating diagrams are mutants of each other by their signed planar graphs.     

An existence theorem for planar maps

The main result of this paper is a theorem concerning possible cubic maps on the plane or sphere. The dual approach of spherical triangulations will be used. Every such triangulation must contain a vertex of degree less than 8 and other than 5, or else contain one of a list of 5 configurations. Due to the occurrence of these five configurations in four color reduction arguments, this implies that a minimal five color map must have at least one face with six or seven neighbors. The theorem is given in Heesch [1, p. 129 ff.], but the proof here is much shorter, due to a modification of Heesch's principle of “discharging”. The principle itself is obtained by exploiting the Euler formula, and should be compared with the theory of Euler contributions as developed by Ore in [3], and by Ore and Stemple [4].