Almost Disjoint Triangles in 3-Space

   Abstract. Two triangles are called almost disjoint if they are either disjoint or their intersection consists of one common vertex. Let f(n) denote the maximum number of pairwise almost disjoint triangles that can be found on some vertex set of n points in 3-space. Here we prove that f(n)=Ω(n ^3/2 ) .     

Adjoining Almost Disjoint Permutations

We show that it is consistent with ZFC + ¬CH that there is a maxima a most disjoint permutation family A ⊆ Sym(ℕ)such that A is a proper subset of an eventually different family E ⊆ ^ℕℕ and |A| < |E|. We also ask severa questions in this area.     

Coincidence Site Modules in 3-Space

The coincidence site lattice (CSL) problem and its generalization to ℤ-modules in Euclidean 3-space is revisited, and various results and conjectures are proved in a unified way, by using maximal orders in quaternion algebras of class number 1 over real algebraic number fields.     

Rigid Ball-Polyhedra in Euclidean 3-Space

A ball-polyhedron is the intersection with non-empty interior of finitely many (closed) unit balls in Euclidean $3$ -space. One can represent the boundary of a ball-polyhedron as the union of vertices, edges, and faces defined in a rather natural way. A ball-polyhedron is called a simple ball-polyhedron if at every vertex exactly three edges meet. Moreover, a ball-polyhedron is called a standard ball-polyhedron if its vertex–edge–face structure is a lattice (with respect to containment). To each edge of a ball-polyhedron, one can assign an inner dihedral angle and say that the given ball-polyhedron is locally rigid with respect to its inner dihedral angles if the vertex–edge–face structure of the ball-polyhedron and its inner dihedral angles determine the ball-polyhedron up to congruence locally. The main result of this paper is a Cauchy-type rigidity theorem for ball-polyhedra stating that any simple and standard ball-polyhedron is locally rigid with respect to its inner dihedral angles.     

Extended Rectifying Curves in Minkowski 3-Space

In this paper, we consider nonhelical rectifying curves using the any orthonormal moving frame in Minkowski 3?space. Then, we give some relations between nonhelical rectifying curves and their Darboux vectors. Also, we demonstrate that the modified Darboux vectors of the curves are rectifying curves. So, this study is a general expression for the known rectifying curves. In the last section, we give some examples and draw their pictures.     

Triangles in 3-connected matroids

A collection F of 3-connected matroids is triangle-rounded if, whenever M is a 3-connected matroid having a minor in F, and T is a 3-element circuit of M, then M has a minor which uses T and is isomorphic to a member of F. An efficient theorem for testing a collection of matroids for this property is presented. This test is used to obtain several results including the following extension of a result of Asano, Nishizeki, and Seymour. Let T be a 3-element circuit of a 3-connected binary nonregular matroid M with at least eight elements. Then M has a minor using T that is isomorphic to S₈ or the generalized parallel connection across T of F₇ and M(K₄).     

Linkless and Flat Embeddings in 3-Space

We consider piecewise linear embeddings of graphs in 3-space ℝ³. Such an embedding is linkless if every pair of disjoint cycles forms a trivial link (in the sense of knot theory). Robertson, Seymour and Thomas (J. Comb. Theory, Ser. B 64:185–227, 1995) showed that a graph has a linkless embedding in ℝ³ if and only if it does not contain as a minor any of seven graphs in Petersen’s family (graphs obtained from K ₆ by a series of YΔ and ΔY operations). They also showed that a graph is linklessly embeddable in ℝ³ if and only if it admits a flat embedding into ℝ³, i.e. an embedding such that for every cycle C of G there exists a closed 2-disk D⊆ℝ³ with D∩G=∂D=C. Clearly, every flat embedding is linkless, but the converse is not true. We consider the following algorithmic problem associated with embeddings in ℝ³:     

Boundaries of Flat Compact Surfaces in 3-Space

This paper deals with the problem `which knots or links in3-space bound flat (immersed) compact surfaces?' In aforthcoming paper by the author, it is proven that any simple closedspace curve can be deformed until it bounds a flat orientable compact(Seifert) surface. The main results of this paper are that there existknots that do not bound any flat compact surfaces. The lower bound oftotal curvature of a knot bounding an orientable nonnegatively curvedcompact surface can, for varying knot types, be arbitrarily much greaterthan the infimum of curvature needed for the knot to have its knot type.The number of 3-singular points (points of zero curvatureor if not then of zero torsion) on the boundary of a flat immersedcompact surface is greater than or equal to twice the absolute value ofthe Euler characteristic of the surface. A set of necessary and, in aweakened sense, sufficient conditions for a knot or link to be what wecall a generic boundary of a flat immersed compact surface withoutplanar regions is given.     

Symmetries of Real Hypersurfaces in Complex 3-Space

The main result of the paper consists in the proof of the fact that for any germ of a real analytic hypersurface in complex 3-space the following alternative (dimension conjecture) takes place: either the dimension of the group of holomorphic symmetries of the germ is at most the dimension of that of a nondegenerate hyperquadric (the latter equals 15), or the group is infinite-dimensional. We also discuss mistakes found in A. Ershova’s paper.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 171–179.Original Russian Text Copyright © 2005 by V. K. Beloshapka.     

The Sub-Parabolic Lines in the Minkowski 3-Space

Results in Mathematics - In this paper, we investigate the focal surfaces obtained by the normal rectilinear congruence in the Minkowski 3-space. The sub-parabolic points and the ridge points are...     

A New Packing Density Bound in 3-Space

It is shown that every compact convex set K which is centrally symmetric and has a non-empty interior admits a lattice packing of Euclidean 3-space with density greater than or equal to 0.53835.... This is an improvement of the result in [8], which achieved a bound of 0.46421.... Minkowski combinations and the Brunn-Minkowski inequality are used in conjunction with the construction in [8] to achieve a better result.     

A Note on Minimal Surfaces in Euclidean 3-Space

In this note, a construction of minimal surfaces in Euclidean 3-space is given. By using the product of Weierstrass data of two known minimal surfaces, one gets a new Weierstrass data and a corresponding minimal surface from the Weierstrass representation.     

Spinor Bishop Equations of Curves in Euclidean 3-Space

In this paper, we study spinor Bishop equations of curves in ${\mathbb{E}^3}$ . We research the spinor formulations of curves according to Bishop frames in ${\mathbb{E}^3}$ . Also, the relations between spinor formulations of Bishop frames and Frenet frame are expressed.     

On Helicoidal Surfaces in a Conformally Flat 3-Space

In this paper, we discuss the problem of finding explicit parametrizations for the helicoidal surfaces in a conformally flat 3-space \(\mathbb {E}^3_F\) with prescribed extrinsic curvature or mean curvature given by smooth functions. Also, we give examples for helicoidal surfaces with some extrinsic curvature and mean curvature functions in \(\mathbb {E}^3_F\).     

Large Sets of Mutually Almost Disjoint Steiner Triple Systems Not From Steiner Quadruple Systems

We construct what we believe to be the first examples of large sets of MAD STS( ) which are not obtained from Steiner quadruple systems     

关于3维Minkowski空间中类空曲面的若干结果

在本文，我们证明２维黎曼流形能实现为Ｌ３中的类空极大曲面的充要条件是相应的Ｒｉｃｃｉ条件成立，此外还确定了Ｌ３中其平均曲率向量ｈ满足条件△ｈ＝λｈ（λ∈Ｒ）的类空曲面     

关于三维Minkowski空间中直线汇的一些注记

本文研究了三维Minkowski空间中直线汇的一些性质,特别是关于类时线汇的性质．讨论了线汇基本元素的存在性,并证明了关于三维Minkowski空间中类时线汇的配分参数的一个结果,推广了苏步青1927年的—个成果．