A note on commutative semifield planes

2018 ◽  
Vol 18 (1) ◽  
pp. 115-118
Author(s):  
Yue Zhou
Keyword(s):  
Projective Plane ◽  
Prime Power ◽  
Semifield Plane ◽  
Planar Function ◽  
Semifield Planes

AbstractLetqbe an odd prime power. We prove that a planar functionffrom 𝔽qto itself can be written as an affine Dembowski–Ostrom polynomial if and only if the projective plane derived fromfis a commutative semifield plane.

Reidemeister Projective Planes

1968 ◽  
Vol 20 ◽  
pp. 1459-1464
Author(s):  
Michael J. Kallaher
Keyword(s):  
Projective Plane ◽  
Prime Power ◽  
Projective Planes ◽  
Ternary Ring ◽  
The Third

By a Reidemeister plane we mean a projective plane having the property that every ternary ring coordinatizing it has associative addition. Finite Reidemeister planes have been investigated by Gleason (2), Liineburg (6), and Kegel and Luneburg (4). In the first paper, Gleason proved that if the order of the plane is a prime power, then it is Desarguesian. Luneburg showed that this is still true if the order is not 60. In the third paper, this last restriction is removed. For infinite planes, the only result is the following theorem due to Pickert (7, p. 301).

The Non-Existence of Finite Projective Planes of Order 10

1989 ◽  
Vol 41 (6) ◽  
pp. 1117-1123 ◽  
Author(s):  
C. W. H. Lam ◽  
L. Thiel ◽  
S. Swiercz
Keyword(s):  
Projective Plane ◽  
Prime Power ◽  
Projective Planes ◽  
Latin Squares ◽  
Orthogonal Latin Squares ◽  
Finite Projective Planes

A finite projective plane of order n, with n > 0, is a collection of n2+ n + 1 lines and n2+ n + 1 points such that1. every line contains n + 1 points,2. every point is on n + 1 lines,3. any two distinct lines intersect at exactly one point, and4. any two distinct points lie on exactly one line.It is known that a plane of order n exists if n is a prime power. The first value of n which is not a prime power is 6. Tarry [18] proved in 1900 that a pair of orthogonal latin squares of order 6 does not exist, which by Bose's 1938 result [3] implies that a projective plane of order 6 does not exist.

scholarly journals A Generalization of Some Huang–Johnson Semifields

10.37236/516 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
N. L. Johnson ◽  
Giuseppe Marino ◽  
Olga Polverino ◽  
Rocco Trombetti
Keyword(s):  
Finite Fields ◽  
Prime Power ◽  
Applied Mathematics ◽  
Discrete Math ◽  
Translation Planes ◽  
Type Ii ◽  
Finite Translation ◽  
Semifield Planes

In [H. Huang, N.L. Johnson: Semifield planes of order $8^2$, Discrete Math., 80 (1990)], the authors exhibited seven sporadic semifields of order $2^6$, with left nucleus ${\mathbb F}_{2^3}$ and center ${\mathbb F}_2$. Following the notation of that paper, these examples are referred as the Huang–Johnson semifields of type $II$, $III$, $IV$, $V$, $VI$, $VII$ and $VIII$. In [N. L. Johnson, V. Jha, M. Biliotti: Handbook of Finite Translation Planes, Pure and Applied Mathematics, Taylor Books, 2007], the question whether these semifields are contained in larger families, rather then sporadic, is posed. In this paper, we first prove that the Huang–Johnson semifield of type $VI$ is isotopic to a cyclic semifield, whereas those of types $VII$ and $VIII$ belong to infinite families recently constructed in [N.L. Johnson, G. Marino, O. Polverino, R. Trombetti: Semifields of order $q^6$ with left nucleus ${\mathbb F}_{q^3}$ and center ${\mathbb F}_q$, Finite Fields Appl., 14 (2008)] and [G.L. Ebert, G. Marino, O. Polverino, R. Trombetti: Infinite families of new semifields, Combinatorica, 6 (2009)]. Then, Huang–Johnson semifields of type $II$ and $III$ are extended to new infinite families of semifields of order $q^6$, existing for every prime power $q$.

Gilles Deleuze's Luminous Philosophy

2020 ◽  
Author(s):  
Hanjo Berressem
Keyword(s):  
Projective Plane ◽  
Visual Arts ◽  
Whole Body ◽  
Literary Studies ◽  
Chronological Order ◽  
Real Projective Plane ◽  
Cinema Studies ◽  
The Real

Providing a comprehensive reading of Deleuzian philosophy, Gilles Deleuze’s Luminous Philosophy argues that this philosophy’s most consistent conceptual spine and figure of thought is its inherent luminism. When Deleuze notes in Cinema 1 that ‘the plane of immanence is entirely made up of light’, he ties this philosophical luminism directly to the notion of the complementarity of the photon in its aspects of both particle and wave. Engaging, in chronological order, the whole body and range of Deleuze’s and Deleuze and Guattari’s writing, the book traces the ‘line of light’ that runs through Deleuze’s work, and it considers the implications of Deleuze’s luminism for the fields of literary studies, historical studies, the visual arts and cinema studies. It contours Deleuze’s luminism both against recent studies that promote a ‘dark Deleuze’ and against the prevalent view that Deleuzian philosophy is a philosophy of difference. Instead, it argues, it is a philosophy of the complementarity of difference and diversity, considered as two reciprocally determining fields that are, in Deleuze’s view, formally distinct but ontologically one. The book, which is the companion volume toFélix Guattari’s Schizoanalytic Ecology, argues that the ‘real projective plane’ is the ‘surface of thought’ of Deleuze’s philosophical luminism.

scholarly journals Group-labeled light dual multinets in the projective plane

2018 ◽  
Vol 341 (8) ◽  
pp. 2121-2130 ◽  
Author(s):  
Gábor Korchmáros ◽  
Gábor P. Nagy
Keyword(s):  
Projective Plane

scholarly journals On 3-syzygy and unexpected plane curves

2021 ◽  
Author(s):  
Grzegorz Malara ◽  
Piotr Pokora ◽  
Halszka Tutaj-Gasińska
Keyword(s):  
Projective Plane ◽  
Plane Curves ◽  
Complex Projective Plane ◽  
Geometric Properties ◽  
Complex Projective

AbstractIn this note we study curves (arrangements) in the complex projective plane which can be considered as generalizations of free curves. We construct families of arrangements which are nearly free and possess interesting geometric properties. More generally, we study 3-syzygy curve arrangements and we present examples that admit unexpected curves.

scholarly journals An infinite family of Williamson matrices

1977 ◽  
Vol 24 (2) ◽  
pp. 252-256 ◽  
Author(s):  
Edward Spence
Keyword(s):  
Prime Power ◽  
Hadamard Matrix ◽  
Infinite Family ◽  
Williamson Matrices

AbstractIn this paper the following result is proved. Suppose there exists a C-matrix of order n + 1. Then if n≡1 (mod 4) there exists a Hadamard matrix of order 2nr(n + 1), while if n≡3 (mod 4) there exists a Hadamard matrix of order nr(n + 1) for all r ≧0. If n≡1 (mod 4) is a prime power, the method is adapted to prove the existence of a Hadamard matrix of the Williamson type, of order 2nr(n + 1), for all r ≧0.

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