New upper bounds on the maximal size of an arc in a projective Hjelmslev plane

2020 ◽  
Author(s):  
Thomas Honold ◽  
Michael Kiermaier ◽  
Ivan Landjev
Keyword(s):  
Upper Bounds ◽  
Hjelmslev Plane ◽  
Projective Hjelmslev Plane ◽  
Maximal Size

Locally Compact Hjelmslev Planes and Rings

1981 ◽  
Vol 33 (4) ◽  
pp. 988-1021 ◽  
Author(s):  
J. W. Lorimer
Keyword(s):  
Projective Planes ◽  
Equivalence Relations ◽  
Topological Spaces ◽  
Hjelmslev Plane ◽  
Projective Hjelmslev Plane ◽  
Quotient Spaces ◽  
Continuous Maps ◽  
Topological Plane ◽  
Line Point ◽  
Canonical Image

Affine and projective Hjelmslev planes are generalizations of ordinary affine and projective planes where two points (lines) may be joined by (may intersect in) more than one line (point). The elements involved in multiple joinings or intersections are neighbours, and the neighbour relations on points respectively lines are equivalence relations whose quotient spaces define an ordinary affine or projective plane called the canonical image of the Hjelmslev plane. An affine or projective Hjelmslev plane is a topological plane (briefly a TH-plane and specifically a TAH-plane respectively a TPH-plane) if its point and line sets are topological spaces so that the joining of non-neighbouring points, the intersection of non-neighbouring lines and (in the affine case) parallelism are continuous maps, and the neighbour relations are closed.In this paper we continue our investigation of such planes initiated by the author in [38] and [39].

Dual Numbers and Topological Hjelmslev Planes

1983 ◽  
Vol 26 (3) ◽  
pp. 297-302 ◽  
Author(s):  
J. W. Lorimer
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Dual Numbers ◽  
Hjelmslev Plane ◽  
Product Topology ◽  
Topological Dimension ◽  
Locally Compact ◽  
Projective Hjelmslev Plane ◽  
Zero Divisors ◽  
Unique Maximal Ideal

AbstractIn 1929 J. Hjelmslev introduced a geometry over the dual numbers ℝ+tℝ with t2 = Q. The dual numbers form a Hjelmslev ring, that is a local ring whose (unique) maximal ideal is equal to the set of 2 sided zero divisors and whose ideals are totally ordered by inclusion. This paper first shows that if we endow the dual numbers with the product topology of ℝ2, then we obtain the only locally compact connected hausdorfT topological Hjelmslev ring of topological dimension two. From this fact we establish that Hjelmslev's original geometry, suitably topologized, is the only locally compact connected hausdorfr topological desarguesian projective Hjelmslev plane to topological dimension four.

scholarly journals On projective Hjelmslev planes of level n

1989 ◽  
Vol 31 (3) ◽  
pp. 257-261 ◽  
Author(s):  
G. Hanssens ◽  
H. van Maldeghem
Keyword(s):  
Existence Theorem ◽  
Inverse Limits ◽  
Hjelmslev Plane ◽  
Projective Hjelmslev Plane ◽  
Equivalent Definition ◽  
Triangle Building ◽  
Infinite Case ◽  
Definition Of

In this paper, we establish a new (but equivalent) definition of projective Hjelmslev planes of level n. This shows that the nth floor of a triangle building is a projective Hjelmslev plane of level n (a result already announced in [9], but left unproved). This will allow us to characterize Artmann-sequences by means of their inverse limits and to construct new ones. We also deduce a new existence theorem for level n projective Hjelmslev planes. All results hold in the finite as well as in the infinite case.

Embedding Right Chain Rings in Chain Rings

1978 ◽  
Vol 30 (5) ◽  
pp. 1079-1086 ◽  
Author(s):  
H. H. Brungs ◽  
G. Törner
Keyword(s):  
Maximal Ideal ◽  
Coordinate Ring ◽  
Hjelmslev Plane ◽  
Chain Ring ◽  
Projective Hjelmslev Plane ◽  
Algebraic Version ◽  
Zero Divisors ◽  
Starting Point ◽  
Affine Hjelmslev Plane ◽  
A Chain

The following problem was the starting point for this investigation: Can every desarguesian affine Hjelmslev plane be embedded into a desarguesian projective Hjelmslev plane [8]? An affine Hjelmslev plane is called desarguesian if it can be coordinatized by a right chain ring R with a maximal ideal J(R) consisting of two-sided zero divisors. A projective Hjemslev plane is called desarguesian if the coordinate ring is in addition a left chain ring, i.e. a chain ring. This leads to the algebraic version of the above problem, namely the embedding of right chain rings into suitable chain rings. We can prove the following result.

scholarly journals A Complexity Analysis of Functional Interpretations

2003 ◽  
Vol 10 (12) ◽  
Author(s):  
Mircea-Dan Hernest ◽  
Ulrich Kohlenbach
Keyword(s):  
Quantitative Analysis ◽  
Time Complexity ◽  
Complexity Analysis ◽  
Upper Bounds ◽  
Weak Base ◽  
Functional Interpretation ◽  
Maximal Size ◽  
Negative Translation

We give a quantitative analysis of Gödel's functional interpretation and its monotone variant. The two have been used for the extraction of programs and numerical bounds as well as for conservation results. They apply both to (semi-)intuitionistic as well as (combined with negative translation) classical proofs. The proofs may be formalized in systems ranging from weak base systems to arithmetic and analysis (and numerous fragments of these). We give upper bounds in basic proof data on the depth, size, maximal type degree and maximal type arity of the extracted terms as well as on the depth of the verifying proof. In all cases terms of size linear in the size of the proof at input can be extracted and the corresponding extraction algorithms have cubic worst-time complexity. The verifying proofs have depth linear in the depth of the proof at input and the maximal size of a formula of this proof.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

Energy of spherical fuzzy graphs

2020 ◽  
pp. 321-332
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Fuzzy Graph ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

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