A symmetry theorem for Laguerre planes

A 4-point Pascal theorem on an oval in a protective plane led to the symmetry theorem in the Minkowski plane, and this relation was used by the author in [2,3] to prove that the symmetry theorem is equivalent to Miquel's theorem and that it implies the tangency theorem (Berührsatz). The same special Pascal theorem now leads to an incidence theorem, , for the Laguerre plane, which is again equivalent to Miquel's theorem. The configuration of contains 6 points and 5 circles and is thus simpler than that of Miquel, although it does not have the intuitive symmetry properties of the corresponding configuration in the Minkowski plane.     

Schliessungssätze in Laguerre-Ebenen

For Laguerre planes Artzy [1] showed that a 4-point Pascal theorem on an oval leads to a configuration , consisting of six points and five regular circles, which is equivalent Miquel's theorem. We represent some similar incidence assumptions which are again equivalent to in each Laguerre plane. Besides, a uniform denotation to characterize different kinds of miquelian theorems in Benz planes is suggested.

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Gewidmet Herrn Professor Benz zum 60. Geburtstag     

A pascal theorem applied to Minkowski geometry

If on an oval in a projective plane a 4-point Pascal theorem, , with fixed points U and V holds, then the oval is {(x,y) ¦xy=c} (O) (), with c O, in some Hall coordinatization. If for every 3 distinct points P, Q, R (not on UV; neither U nor V collinear with two of P, Q, R) there is through them a certain point set satisfying an extended version of , then all these sets together with all lines not through U or V form the circles of a plane Minkowski (= pseudoeuclidean) geometry over a commutative field. may be expressed in terms of Minkowski geometry. Together with incidence axioms derived from the protective incidence axioms, the Minkowski version of characterizes the plane Minkowski geometry over a commutative field and is thus equivalent to Miquel's theorem.     

A Bernstein theorem for complete spacelike constant mean curvature hypersurfaces in Minkowski space

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As an application, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hypersurface must be linear. This result extends the previous theorems obtained by B. Palmer [Pa] and Y.L. Xin [Xin1] where they assume that the image of the Gauss map is bounded. We also prove a Bernstein theorem for spacelike complete surfaces with parallel mean curvature vector in four-dimensional spaces. Received July 4, 1997 / Accepted October 9, 1997     

A finitization of Littlewood's Tauberian theorem and an application in Tauberian remainder theory

In this paper we study Littlewood's Tauberian theorem from a proof theoretic perspective. We first use the Dialectica interpretation to produce an equivalent, finitary formulation of the theorem, and then carry out an analysis of Wielandt's proof to extract concrete witnessing terms. We argue that our finitization can be viewed as a generalized Tauberian remainder theorem, and we instantiate it to produce two concrete remainder theorems as a corollary, in terms of rates of convergence and rates metastability, respectively. We rederive the standard remainder estimate for Littlewood's theorem as a special case of the former.     

On a theorem about projectivities

The main result of this paper is a theorem about projectivities in then-dimensional complex projective spaceP ⁿ (n 2). Puttingn = 2, we showed in [3] that the theorem of Desargues inP ⁿ is a special case of this theorem. And not only the theorem of Desargues but also the converse of the theorem of Pascal, the theorem of Pappus-Pascal, the theorem of Miquel, the Newton line, the Brocard points and a lot of lesser known results in the projective, the affine and the Euchdean plane were obtained from this theorem as special cases without any further proof. Many extensions of classical theorems in the projective, affine and Euclidean plane to higher dimensions can be found in the literature and probably some of these are special cases of this theorem inP ⁿ. We only give a few examples and leave it as an open problem which other special cases could be found.     

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.     

A case study in formalizing projective geometry in Coq: Desargues theorem

Formalizing geometry theorems in a proof assistant like Coq is challenging. As emphasized in the literature, the non-degeneracy conditions lead to long technical proofs. In addition, when considering higher-dimensions, the amount of incidence relations (e.g. point–line, point–plane, line–plane) induce numerous technical lemmas. In this article, we investigate formalizing projective plane geometry as well as projective space geometry. We mainly focus on one of the fundamental properties of the projective space, namely Desargues property. We formally prove that it is independent of projective plane geometry axioms but can be derived from Pappus property in a two-dimensional setting. Regarding at least three-dimensional projective geometry, we present an original approach based on the notion of rank which allows to describe incidence and non-incidence relations such as equality, collinearity and coplanarity homogeneously. This approach allows to carry out proofs in a more systematic way and was successfully used to fairly easily formalize Desargues theorem in Coq. This illustrates the power and efficiency of our approach (using only ranks) to prove properties of the projective space.     

The divergence theorem and the Laplacian in Minkowski space

Let (X, B) be a Minkowski space (finite-dimensional Banach space) with unit ball B. Using a Minkowski definition of unit normal to a hypersurface, a Minkowski analogue of Euclidean divergence is defined. We show that the divergence theorem holds. Using the Minkowski divergence, a Minkowski Laplacian is defined. We prove that this Laplacian is a second-order, constant-coefficient, elliptic, differential operator. Furthermore, the symbol of this Laplacian is computed and used to associate a natural Euclidean structure with (X, B).Supported, in part, by NSERC Operating Grant #4066.     

On Napoleon's theorem in the isotropic plane]{On Napoleon's theorem in the isotropic plane

Summary Napoleon's original theorem refers to arbitrary triangles in the Euclidean plane. If equilateral triangles are externally erected on the sides of a given triangle, then their three corresponding circumcenters form an equilateral triangle. We present some analogous theorems and related statements for the isotropic (Galilean) plane.     

Eight and seven point degenerations of Miquel's Theorem in Laguerre planes

For M?bius planes Schaeffer [9] has proved that all seven point degenerations of Miquel's Theorem characterize miquelian M?bius planes. For Laguerre planes we have several degenerations of Miquel's Theorem with eight and seven points. We prove that all except one of these degenerations characterize miquelian Laguerre planes. The remaining degeneration characterizes elation Laguerre planes. Received 14 September 2001; revised 1 September 2002.     

On a theorem of Erdös and Szekeres

In this note we generalize a theorem of Erdös and Szekeres, which states that every sequence of real numbers of length n² + 1 has a monotone subsequence of length n + 1, for points in certain metric spaces (R^k, d), where d is a Minkowski metric. Three theorems are proved concerning preassigned numbers of points which must lie on the same geodesic of the space, the last of which characterizes the class of Minkowski spaces under discussion.     

Convexity in Topological Affine Planes

We extend to topological affine planes the standard theorems of convexity, among them the separation theorem, the anti-exchange theorem, Radon's, Helly's, Caratheodory's, and Kirchberger's theorems, and the Minkowski theorem on extreme points. In a few cases the proofs are obtained by adapting proofs of the original results in the Euclidean plane; in others it is necessary to devise new proofs that are valid in the more general setting considered here.     

Alternatives to the Halpern-Läuchli theorem

In this paper, it is shown that the fact that the BPI holds in the Cohen's symmetric model can be used as an equal substitute for the Halpern-Läuchli theorem. Also, some alternatives to the Halpern-Läuchli theorem in the form of absoluteness theorems for a certain class of statements are consequences¹ of such consistency results. The article also contains a new proof of the Halpern-Läuchli theorem.     

A fixed point theorem for the infinite-dimensional simplex

We define the infinite-dimensional simplex to be the closure of the convex hull of the standard basis vectors in R^∞, and prove that this space has the fixed point property: any continuous function from the space into itself has a fixed point. Our proof is constructive, in the sense that it can be used to find an approximate fixed point; the proof relies on elementary analysis and Sperner's lemma. The fixed point theorem is shown to imply Schauder's fixed point theorem on infinite-dimensional compact convex subsets of normed spaces.     

内函数逼近定理及上溢原理的推广及应用

在κ-饱和的非标准模型中,首先推广了内函数逼近定理,并用这一推广定理证明了著名的A sco li定理;其次将上(下)溢原理推广到一般的定向集上,并证明了拓扑空间中单子的一些性质,给出了无穷小延伸定理的一个简单证明.     

Uniqueness theorem for locally antipodal Delaunay sets

We prove theorems on locally antipodal Delaunay sets. The main result is the proof of a uniqueness theorem for locally antipodal Delaunay sets with a given 2R-cluster. This theorem implies, in particular, a new proof of a theorem stating that a locally antipodal Delaunay set all of whose 2R-clusters are equivalent is a regular system, i.e., a Delaunay set on which a crystallographic group acts transitively.     

Omitting types theorem in hybrid dynamic first-order logic with rigid symbols

In the present contribution, we prove an Omitting Types Theorem (OTT) for an arbitrary fragment of hybrid dynamic first-order logic with rigid symbols (i.e. symbols with fixed interpretations across worlds) closed under negation and retrieve. The logical framework can be regarded as a parameter and it is instantiated by some well-known hybrid and/or dynamic logics from the literature. We develop a forcing technique and then we study a forcing property based on local satisfiability, which lead to a refined proof of the OTT. For uncountable signatures, the result requires compactness, while for countable signatures, compactness is not necessary. We apply the OTT to obtain upwards and downwards Löwenheim-Skolem theorems for our logic, as well as a completeness theorem for its constructor-based variant.     

Phelps' lemma, Danes' drop theorem and Ekeland's principle in locally convex spaces

A generalization of Phelps' lemma to locally convex spaces is proven, applying its well-known Banach space version. We show the equivalence of this theorem, Ekeland's principle and Danes' drop theorem in locally convex spaces to their Banach space counterparts and to a Pareto efficiency theorem due to Isac. This solves a problem, concerning the drop theorem, proposed by G. Isac in 1997.

We show that a different formulation of Ekeland's principle in locally convex spaces, using a family of topology generating seminorms as perturbation functions rather than a single (in general discontinuous) Minkowski functional, turns out to be equivalent to the original version.

     

A Bernstein theorem for complete spacelike hypersurfaces of constant mean curvature in Minkowski space

In this paper we prove a general Bernstein theorem on the complete spacelike constant mean curvature hypersurfaces in Minkowski space. The result generalizes the previous result of Cao-Shen-Zhu (1998) and Xin (1991). The proof again uses the fact that the Gauss map of a constant mean curvature hypersurface is harmonic, which was proved by K. T. Milnor (1983), and the maximum principle of S. T. Yau (1975).