On a geometric statement of Ramsey type

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Alexander Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Set Theory ◽  
Ramsey Type ◽  
Straight Lines

Abstract A simple geometric assertion of Ramsey type, concerning families of straight lines in the Euclidean space R 3 \mathbb{R}^{3} , is formulated, and it is shown that the assertion turns out to be undecidable within the framework of ZFC set theory.

Unions of rectifiable curves in Euclidean space and the covering number of the meagre ideal

1999 ◽  
Vol 64 (2) ◽  
pp. 701-726 ◽  
Author(s):  
Juris Steprāns
Keyword(s):  
Euclidean Space ◽  
Banach Spaces ◽  
Set Theory ◽  
Metric Space ◽  
Covering Number ◽  
Cardinal Invariant ◽  
Rectifiable Curves ◽  
Straight Lines

AbstractTo any metric space it is possible to associate the cardinal invariant corresponding to the least number of rectifiable curves in the space whose union is not meagre. It is shown that this invariant can vary with the metric space considered, even when restricted to the class of convex subspaces of separable Banach spaces. As a corollary it is obtained that it is consistent with set theory that any set of reals of size ℵ1 is meagre yet there are ℵ1 rectifiable curves in ℝ3 whose union is not meagre. The consistency of this statement when the phrase “rectifiable curves” is replaced by “straight lines” remains open.

Linear Functions with Two Points of Intersection?

1995 ◽  
Vol 88 (5) ◽  
pp. 376-378
Author(s):  
Michael A. Contino
Keyword(s):  
Euclidean Space ◽  
Graphing Calculator ◽  
Linear Functions ◽  
First Year ◽  
Systems Of Equations ◽  
College Mathematics ◽  
Algebra Students ◽  
Straight Lines

Of course two straight lines in Euclidean space cannot intersect in more than one point unless they are the same line and intersect everywhere—or can they? Follow this problem on the graphing calculator, however, and the surprising twist that gives this article its name will be seen. The material covered should be readily accessible to first-year-algebra students who have studied systems of equations, but it also contains valuable lessons for college mathematics professors who have been easily deceived by its apparent simplicity and familiarity.

scholarly journals Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-Space

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1256
Author(s):  
Rafael López ◽  
Željka Milin Šipuš ◽  
Ljiljana Primorac Gajčić ◽  
Ivana Protrka
Keyword(s):  
Euclidean Space ◽  
Euclidean Plane ◽  
Reconstruction Process ◽  
Null Curve ◽  
Straight Line ◽  
Null Curves ◽  
Straight Lines

In this paper, we analyze involutes of pseudo-null curves in Lorentz–Minkowski 3-space. Pseudo-null curves are spacelike curves with null principal normals, and their involutes can be defined analogously as for the Euclidean curves, but they exhibit properties that cannot occur in Euclidean space. The first result of the paper is that the involutes of pseudo-null curves are null curves, more precisely, null straight lines. Furthermore, a method of reconstruction of a pseudo-null curve from a given null straight line as its involute is provided. Such a reconstruction process in Euclidean plane generates an evolute of a curve, however it cannot be applied to a straight line. In the case presented, the process is additionally affected by a choice of different null frames that every null curve allows (in this case, a null straight line). Nevertheless, we proved that for different null frames, the obtained pseudo-null curves are congruent. Examples that verify presented results are also given.

scholarly journals Borsuk and Ramsey Type Questions in Euclidean Space

2018 ◽  
pp. 259-277
Author(s):  
Peter Frankl ◽  
János Pach ◽  
Christian Reiher ◽  
Vojtěch Rödl
Keyword(s):  
Euclidean Space ◽  
Ramsey Type

scholarly journals О восстановлении тензорного поля второго ранга с нулевым следом по неполным данным

2020 ◽  
pp. 38-47
Author(s):  
Zyaudin Medzhidov
Keyword(s):  
Euclidean Space ◽  
Displacement Field ◽  
Tensor Field ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Rank 2 ◽  
Straight Lines ◽  
Integral Data ◽  
Linear Integrals

An algorithm for the complete reconstruction of a tensor field of rank 2 in three-dimensional Euclidean space on incomplete integral data is constructed. The solenoid part of the field is constructed using linear integrals on straight lines that intersect a curve at infinity, and the displacement field is constructed using the traceless normal ray integrals.

scholarly journals Independence and consistency proofs in quadratic form theory

1991 ◽  
Vol 56 (4) ◽  
pp. 1195-1211 ◽  
Author(s):  
James E. Baumgartner ◽  
Otmar Spinas
Keyword(s):  
Quadratic Form ◽  
Set Theory ◽  
Orthogonal Group ◽  
Continuum Hypothesis ◽  
Infinite Dimension ◽  
Form Theory ◽  
Countable Dimension ◽  
Straight Lines ◽  
Consistency Proofs ◽  
The Continuum

We consider the following properties of uncountable-dimensional quadratic spaces (E, Φ):(*) For all subspaces U ⊆ E of infinite dimension: dim U˔ < dim E.(**) For all subspaces U ⊆ E of infinite dimension: dim U˔ < ℵ0.Spaces of countable dimension are the orthogonal sum of straight lines and planes, so they cannot have (*), but (**) is trivially satisfied.These properties have been considered first in [G/O] in the process of investigating the orthogonal group of quadratic spaces. It has been shown there (in ZFC) that over arbitrary uncountable fields (**)-spaces of uncountable dimension exist.In [B/G], (**)-spaces of dimension ℵ1 (so (*) = (**)) have been constructed over arbitrary finite or countable fields. But this could be done only under the assumption that the continuum hypothesis (CH) holds in the underlying set theory.

scholarly journals On some properties of normally-inflective complexes with simple inflective center in $E_3$

2021 ◽  
pp. 112
Author(s):  
Ye.N. Ishchenko
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Parametric Families ◽  
Straight Lines ◽  
Arbitrary Curve

In the paper, we consider the special degenerate class of mormally-inflective complexes with simple inflective center in three-dimensional Euclidean space $E_3$. We prove that to construct this class of complexes one should take an arbitrary curve and draw sheaf of straight lines through each point of this curve. For arbitrary normally-inflective complex with simple inflective center we establish that such complex is fibered into two one-parametric families of congruences.

scholarly journals Ramsey Theory Applications

10.37236/34 ◽  
2004 ◽  
Vol 1000 ◽  
Author(s):  
Vera Rosta
Keyword(s):  
Information Theory ◽  
Number Theory ◽  
Computer Science ◽  
Ergodic Theory ◽  
Set Theory ◽  
Ramsey Theory ◽  
Theoretical Computer Science ◽  
Theoretical Computer ◽  
Ramsey Type

There are many interesting applications of Ramsey theory, these include results in number theory, algebra, geometry, topology, set theory, logic, ergodic theory, information theory and theoretical computer science. Relations of Ramsey-type theorems to various fields in mathematics are well documented in published books and monographs. The main objective of this survey is to list applications mostly in theoretical computer science of the last two decades not contained in these.

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