scholarly journals On Distance in Some Finite Planes and Graphs Arising from Those Planes

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
I. Dogan ◽  
A. Akpinar
Keyword(s):  
Euclidean Plane ◽  
Projective Planes ◽  
Central Position ◽  
Finite Affine ◽  
Finite Planes ◽  
Definition Of

In this paper, affine and projective graphs are obtained from affine and projective planes of order p r by accepting a line as a path. Some properties of these affine and projective graphs are investigated. Moreover, a definition of distance is given in the affine and projective planes of order p r and, with the help of this distance definition, the point or points having the most advantageous (central) position in the corresponding graphs are determined, with some examples being given. In addition, the concepts of a circle, ellipse, hyperbola, and parabola, which are well known for the Euclidean plane, are carried over to these finite planes. Finally, the roles of finite affine and projective Klingenberg planes in all the results obtained are considered and their equivalences in graph applications are discussed.

scholarly journals Embedding Cycles in Finite Planes

10.37236/3377 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Felix Lazebnik ◽  
Keith E. Mellinger ◽  
Oscar Vega
Keyword(s):  
Projective Plane ◽  
Affine Plane ◽  
Projective Planes ◽  
Finite Projective Planes ◽  
Finite Affine ◽  
Finite Planes

We define and study embeddings of cycles in finite affine and projective planes. We show that for all $k$, $3\le k\le q^2$,  a $k$-cycle can be embedded in any affine plane of order $q$. We also prove a similar result for finite projective planes: for all $k$, $3\le k\le q^2+q+1$,  a $k$-cycle can be embedded in any projective plane of order $q$.

scholarly journals Blocking and Double Blocking Sets in Finite Planes

10.37236/5717 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Jan De Beule ◽  
Tamás Héger ◽  
Tamás Szőnyi ◽  
Geertrui Van de Voorde
Keyword(s):  
Projective Planes ◽  
Blocking Set ◽  
Blocking Sets ◽  
Desarguesian Plane ◽  
Baer Subplanes ◽  
Desarguesian Projective Planes ◽  
Finite Planes ◽  
Value Sets ◽  
Minimal Blocking ◽  
Desarguesian Affine Plane

In this paper, by using properties of Baer subplanes, we describe the construction of a minimal blocking set in the Hall plane of order $q^2$ of size $q^2+2q+2$ admitting $1-$, $2-$, $3-$, $4-$, $(q+1)-$ and $(q+2)-$secants. As a corollary, we obtain the existence of a minimal blocking set of a non-Desarguesian affine plane of order $q^2$ of size at most $4q^2/3+5q/3$, which is considerably smaller than $2q^2-1$, the Jamison bound for the size of a minimal blocking set in an affine Desarguesian plane of order $q^2$.We also consider particular André planes of order $q$, where $q$ is a power of the prime $p$, and give a construction of a small minimal blocking set which admits a secant line not meeting the blocking set in $1$ mod $p$ points. Furthermore, we elaborate on the connection of this problem with the study of value sets of certain polynomials and with the construction of small double blocking sets in Desarguesian projective planes; in both topics we provide some new results.

Definition of projective planes, examples

2019 ◽  
pp. 1-28
Author(s):  
György Kiss ◽  
Tamás Szőnyi
Keyword(s):  
Projective Planes ◽  
Definition Of

scholarly journals Bundles over Quantum RealWeighted Projective Spaces

Axioms ◽  
2012 ◽  
Vol 1 (2) ◽  
pp. 201-225 ◽  
Author(s):  
Tomasz Brzeziński ◽  
Simon A. Fairfax
Keyword(s):  
Principal Bundle ◽  
Algebraic Approach ◽  
Projective Spaces ◽  
Projective Planes ◽  
Positive Case ◽  
Projective Modules ◽  
Principal Bundles ◽  
Seifert Manifold ◽  
Negative Case ◽  
Definition Of

The algebraic approach to bundles in non-commutative geometry and the definition of quantum real weighted projective spaces are reviewed. Principal U(1)-bundles over quantum real weighted projective spaces are constructed. As the spaces in question fall into two separate classes, the negative or odd class that generalises quantum real projective planes and the positive or even class that generalises the quantum disc, so do the constructed principal bundles. In the negative case the principal bundle is proven to be non-trivial and associated projective modules are described. In the positive case the principal bundles turn out to be trivial, and so all the associated modules are free. It is also shown that the circle (co)actions on the quantum Seifert manifold that define quantum real weighted projective spaces are almost free.

Probleme de la Predication

1998 ◽  
pp. 220-226
Author(s):  
Ezzat Orany
Keyword(s):  
General Problem ◽  
Point Of View ◽  
Central Position ◽  
The Subject ◽  
Definition Of

Some scholars have found the dealing of the problem of predication, or attribution, in the Sophist (251a-e), a "digression," or a treatment of "a trivial question" and "an insignificant example." We propose to reconsider the importance of Plato’s doctrine on the subject from the point of view of the epistemology- ontology relationship in Plato. This leads to a replacement of the passage inside the whole dialogue. Beginning with the definition of the sophist, Plato goes on to treat the "mimetic" art and finds himself confronting a perplexing difficulty: how to understand falsehood, either in thought or in discourse. This is an epistemological difficulty, which raises the central difficulty of how to attribute non-being to being. So, the heart of the matter is the possibility of predication, as Plato states very clearly (238a). The solution arises from the doctrine of the community of species, making possible any attribution of one thing to another. In looking carefully to the dialogue as a whole, we find that the passage 251a-e, dealing with the general problem of predication, occupies a central position, in all meanings, even numerically (between 236e and 264a).

Finite Nets, I. Numerical Invariants

1951 ◽  
Vol 3 ◽  
pp. 94-107 ◽  
Author(s):  
R. H. Bruck
Keyword(s):  
Euclidean Plane ◽  
Latin Squares ◽  
Precise Definition ◽  
Geometrical Object ◽  
Affine Planes ◽  
Orthogonal Latin Squares ◽  
Finite Affine ◽  
Numerical Invariants ◽  
Complete Sets

A finite net N of degree k, order n, is a geometrical object of which the precise definition will be given in §1. The geometrical language of the paper proves convenient, but other terminologies are perhaps more familiar. A finite affine (or Euclidean) plane with n points on each line is simply a net of degree n+ 1, order n (Marshall Hall [1]). A loop of order n is essentially a net of degree 3, order n (Baer [1], Bates [1]). More generally, for , a set of k —2 mutually orthogonal n ⨯ n latin squares may be used to define a net of degree k, order n (and conversely) by paralleling Bose's correspondence (Bose [1]) between affine planes and complete sets of orthogonal latin squares.

scholarly journals The Three Reflections Theorem Revisited

KoG ◽  
2018 ◽  
pp. 41-48
Author(s):  
Gunter Weiss
Keyword(s):  
Projective Geometry ◽  
Euclidean Plane ◽  
Projective Planes ◽  
Common Point ◽  
Point Of View ◽  
Euclidean Case ◽  
Minkowski Metric ◽  
Point Reflection ◽  
Pass Through ◽  
The Given

It is well-known that, in a Euclidean plane, the product of three reflections is again a reflection, iff their axes pass through a common point. For this ``Three reflections Theorem'' (3RT) also non-Euclidean versions exist, see e.g. [4]. This article presents affine versions of it, considering a triplet of skew reflections with axes through a common point. It turns out that the essence of all those cases of 3RT is that the three pairs (axis, reflection direction) of the given (skew) reflections can be observed as an involutoric projectivity. For the Euclidean case and its non-Euclidean counterparts this property is automatically fulfilled. From the projective geometry point of view a (skew) reflection is nothing but a harmonic homology. In the affine situation a reflection is an indirect involutoric transformation, while ``direct'' or ``indirect'' makes no sense in projective planes. A harmonic homology allows an interpretation both, as an axial reflection and as a point reflection. Nevertheless, one might study products of three harmonic homologies, which result in a harmonic homology again. Some special mutual positions of axes and centres of the given homologies lead to elations or even to the identity, too. A consequence of the presented results are further generalisations of the 3RT, e.g. in planes with Minkowski metric, affine or projective 3-space, or in circle geometries.

A purely syntactical definition of confirmation

1943 ◽  
Vol 8 (4) ◽  
pp. 122-143 ◽  
Author(s):  
Carl G. Hempel
Keyword(s):  
Empirical Data ◽  
Empirical Science ◽  
Central Position ◽  
Empirical Hypothesis ◽  
Definition Of ◽  
The Given

The concept of confirmation occupies a central position in the methodology of empirical science. For it is the distinctive characteristic of an empirical hypothesis to be amenable, at least in principle, to a test based on suitable observations or experiments; the empirical data obtained in a test—or, as we shall prefer to say, the observation sentences describing those data—may then either confirm or disconfirm the given hypothesis, or they may be neutral with respect to it. To say that certain observation sentences confirm or disconfirm a hypothesis, does not, of course, generally mean that those observation sentences suffice strictly to prove or to refute the hypothesis in question, but rather that they constitute favorable, or unfavorable, evidence for it; and the term “neutral” is to indicate that the observation sentences are either entirely irrelevant to the hypothesis, or at least insufficient to strengthen or weaken it.

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