Walter Wenzel

Pfaffian forms and D-matroids

Abstract In this paper it is shown that skew-symmetric n × n -matrices with coefficients in a field K correspond via Pfaffian forms in a canonical one-to-one fashion to K -valued maps defined on the power set B ({1,…, n }), which satisfy certain identities. As an application, we describe representability of Δ -matroids by skew-symmetric matrices in terms of these maps. This suggests a definition of orientable and valuated Δ -matroids or, more generally, of Δ -matroids with coefficients which is analogous to the corresponding concept studied in matroid theory.

Tetrahedra are not reduced

Abstract A convex body which does not properly contain a convex body with the same minimal width is said to be reduced. It is not known whether there exist reduced n -polytopes, n ≥ 3. We prove that there is no reduced tetrahedron.

A Characterization of Ordered Sets and Lattices via Betweenness Relations

For a set X with at least 3 elements, we establish a canonical one to one correspondence between all betweenness relations satisfying certain axioms and all pairs of inverse orderings “<” and “>” defined on X for which the corresponding Hasse diagram is connected and all maximal chains contain at least 3 elements. For an ordering “<”, the corresponding betweenness relation B is given by % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!

A Lipschitz condition for the width function of convex bodies in arbitrary Minkowski spaces

B={(x,y,z)in X^3mid x<y<z {rm or }z<y<x}.

Covering and Packing Problems in Lattices Associated with then-Cube

Moreover, by adding one more axiom, we obtain also a one to one correspondence between all pairs of dual lattices and all betweenness relations.

Symmetrization of Closure Operators and Visibility

In this paper we give a new and self-contained proof of the following theorem: Let G denote some finite connected and bipartite graph. If in addition for any two vertices x,y the subgraph induced on those vertices lying on some geodesic from x to y is antipodal, then G is a hypercube of some dimension.

A characterization of convex sets via visibility

Abstract Studying first the Euclidean subcase, we show that the Minkowskian width function of a convex body in an n -dimensional (normed linear or) Minkowski space satisfies a specified Lipschitz condition.

Simplices with congruent k-faces

The coordinates of the cusp are x = −v2ωT/g and y = T 2(g2 − 2v2ω2)/2g. Finally, as an example of a more complicated model to which we can adapt our discussion, we mention the model where air resistance is proportional to velocity. The governing differential equations for x and y are x ′′ + r x ′ = 0 and y′′ + r y′ = −g respectively. The independent variable is T , and the initial conditions are x(0) = y(0) = 0, x ′(0) = v cos(ωt) and y′(0) = v sin(ωt). Of course, these last two are constants, and t is still equal to s − T . After solving, we find that the coordinates of a curve in the photo are given by equations x = v cos(ω(s − T ))(1 − e−r T )/r and y = (g + rv sin(ω(s − T )))(1 − e−r T )/r 2 − gT/r . Figures 3 and 4 illustrate this family of new curves. To facilitate comparison to Figures 1 and 2, we retain their values of v, ω, and s. In all Figures, v = 40 and g = 32. In Figures 1 and 3, ω = 1.0 and s = {2.3, 3.3, 3.8, 4.3}. In Figures 2 and 4, ω = 0.5 and s = {5.3, 5.9, 6.093, 6.3}. For Figures 3 and 4, we use r = 1.