Partitioning Projective Geometries into Caps

In [2] by means of a fairly lengthy argument involving Hermitian varieties it was shown that PG(2n, q2) can be partitioned into (q2n++ 1 + 1)/(q + l)-caps. Moreover, these caps were shown to constitute the “large points” of a PG(2n, q) in a natural way. In [3] a similar argument was used to show that once two disjoint (n – l)-subspaces are removed from PG(2n, q2), the remaining points can be partitioned into (q2n – 1)/(q2 – l)-caps.The purpose of this paper is to give a short proof of the results found in [2], and then use the technique developed to partition PG(2n, q) into (qn + l)-caps for n even and q any prime-power. Moreover, these caps can be treated in a natural way as the “large points” of a PG(n – 1, q).

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Projective Geometries that are Disjoint Unions of Caps

Canadian Journal of Mathematics ◽

10.4153/cjm-1980-099-7 ◽

1980 ◽

Vol 32 (6) ◽

pp. 1299-1305 ◽

Cited By ~ 12

Author(s):

Barbu C. Kestenband

Keyword(s):

Finite Field ◽

Projective Geometry ◽

Prime Power ◽

Disjoint Union ◽

Hermitian Matrices ◽

Incidence Relation ◽

Projective Geometries ◽

Image Position ◽

Natural Way ◽

Square Matrix

We show that any PG(2n, q2) is a disjoint union of (q2n+1 − 1)/ (q − 1) caps, each cap consisting of (q2n+1 + 1)/(q + 1) points. Furthermore, these caps constitute the “large points” of a PG(2n, q), with the incidence relation defined in a natural way.A square matrix H = (hij) over the finite field GF(q2), q a prime power, is said to be Hermitian if hijq = hij for all i, j [1, p. 1161]. In particular, hii ∈ GF(q). If if is Hermitian, so is p(H), where p(x) is any polynomial with coefficients in GF(q).Given a Desarguesian Projective Geometry PG(2n, q2), n > 0, we denote its points by column vectors:All Hermitian matrices in this paper will be 2n + 1 by 2n + 1, n > 0.

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Orthogonal Arrays with Parameters $OA(s^3,s^2+s+1,s,2)$ and 3-Dimensional Projective Geometries

The Electronic Journal of Combinatorics ◽

10.37236/556 ◽

2011 ◽

Vol 18 (1) ◽

Author(s):

Kazuaki Ishii

Keyword(s):

Orthogonal Array ◽

Prime Power ◽

Orthogonal Arrays ◽

3 Dimensional ◽

Projective Geometries ◽

Finite Spaces ◽

Alpha Type

There are many nonisomorphic orthogonal arrays with parameters $OA(s^3,s^2+s+1,s,2)$ although the existence of the arrays yields many restrictions. We denote this by $OA(3,s)$ for simplicity. V. D. Tonchev showed that for even the case of $s=3$, there are at least 68 nonisomorphic orthogonal arrays. The arrays that are constructed by the $n-$dimensional finite spaces have parameters $OA(s^n, (s^n-1)/(s-1),s,2)$. They are called Rao-Hamming type. In this paper we characterize the $OA(3,s)$ of 3-dimensional Rao-Hamming type. We prove several results for a special type of $OA(3,s)$ that satisfies the following condition: For any three rows in the orthogonal array, there exists at least one column, in which the entries of the three rows equal to each other. We call this property $\alpha$-type. We prove the following. (1) An $OA(3,s)$ of $\alpha$-type exists if and only if $s$ is a prime power. (2) $OA(3,s)$s of $\alpha$-type are isomorphic to each other as orthogonal arrays. (3) An $OA(3,s)$ of $\alpha$-type yields $PG(3,s)$. (4) The 3-dimensional Rao-Hamming is an $OA(3,s)$ of $\alpha$-type. (5) A linear $OA(3,s)$ is of $\alpha $-type.

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Hermitian Configurations in Odd-Dimensional Projective Geometries

Canadian Journal of Mathematics ◽

10.4153/cjm-1981-043-7 ◽

1981 ◽

Vol 33 (2) ◽

pp. 500-512 ◽

Cited By ~ 4

Author(s):

Barbu C. Kestenband

Keyword(s):

Disjoint Union ◽

Incidence Relation ◽

Projective Geometries ◽

Natural Way

A t-cap in a geometry is a set of t points no three of which are collinear. A (t, k)-cap is a set of t points, no k + 1 of which are collinear.It has been shown in [3] that any Desarguesian PG(2n, q2) is a disjoint union of (q2n+l – l)/(q – 1) (q2n+l – l)/(q + l)-caps. These caps were obtained as intersections of 2n Hermitian Varieties of a certain kind; the intersection of 2n + 1 such varieties was empty. Furthermore, the caps in question constituted the ‘large points” of a PG(2n, q), with the incidence relation defined in a natural way.It seemed at the time that nothing similar could be said about odd-dimensional projective geometries, if only because |PG(2n – 1, q)| ∤ |PG(2n – l, q2)|.

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On Incomplete Character Sums to a Prime-Power Modulus

Canadian Mathematical Bulletin ◽

10.4153/cmb-1987-037-4 ◽

1987 ◽

Vol 30 (3) ◽

pp. 257-266 ◽

Cited By ~ 1

Author(s):

J. H. H. Chalk

Keyword(s):

Prime Power ◽

Short Proof ◽

Character Sums ◽

Solution Set ◽

Character Sum ◽

Primitive Character ◽

Image Position ◽

Incomplete Character Sums ◽

Main Inequality ◽

Inequality Theorem

AbstractLet x denote a primitive character to a prime-power modulus k = pα. The expected estimatefor the incomplete character sum has been established for r = 1 and 2 by D. A. Burgess and recently, he settled the case r = 3 for all primes p < 3, (cf. [2] for the proof and for references). Here, a short proof of the main inequality (Theorem 2) which leads to this result is presented; the argument being based upon my characterization in [3] of the solution-set of a related congruence.

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Critical Exponents, Colines, and Projective Geometries

Combinatorics Probability Computing ◽

10.1017/s0963548300004326 ◽

2000 ◽

Vol 9 (4) ◽

pp. 355-362 ◽

Cited By ~ 3

Author(s):

JOSEPH P. S. KUNG

Keyword(s):

Positive Integer ◽

Prime Power ◽

Dimensional Subspace ◽

Complete Graphs ◽

Matroid Theory ◽

Primary Interest ◽

Projective Geometries ◽

Critical Problems ◽

Simple Matroid ◽

Geometric Analogue

In [9, p. 469], Oxley made the following conjecture, which is a geometric analogue of a conjecture of Lovász (see [1, p. 290]) about complete graphs.Conjecture 1.1.Let G be a rank-n GF(q)-representable simple matroid with critical exponent n − γ. If, for every coline X in G, c(G/X; q) = c(G; q) − 2 = n − γ − 2, then G is the projective geometry PG(n − 1, q).We shall call the rank n, the critical ‘co-exponent’ γ, and the order q of the field the parameters of Oxley's conjecture. We exhibit several counterexamples to this conjecture. These examples show that, for a given prime power q and a given positive integer γ, Oxley's conjecture holds for only finitely many ranks n. We shall assume familiarity with matroid theory and, in particular, the theory of critical problems. See [6] and [9].A subset C of points of PG(n − 1, q) is a (γ, k)-cordon if, for every k-codimensional subspace X in PG(n − 1, q), the intersection C ∩ X contains a γ-dimensional subspace of PG(n − 1, q). In this paper, our primary interest will be in constructing (γ, 2)-cordons. With straightforward modifications, our methods will also yield (γ, k)-cordons.Complements of counterexamples to Oxley's conjecture are (γ, 2)-cordons.

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INEVITABILITY OF REVIVAL: PRO AND COUNTER ANTINATALISM

Humanitarian actual problems of the humanities and education ◽

10.15507/2078-9823.045.019.201901.080-088 ◽

2019 ◽

Vol 19 (1) ◽

pp. 80-88

Author(s):

Nikolay S. Savkin

Keyword(s):

Social Relations ◽

Systematic Approach ◽

Human Life ◽

Laws Of Nature ◽

Arthur Schopenhauer ◽

Expected Effect ◽

Philosophy Of Life ◽

Active Subject ◽

Over Time ◽

Natural Way

Introduction. Radical pessimism and militant anti-natalism of Arthur Schopenhauer and David Benathar create an optimistic philosophy of life, according to which life is not meaningless. It is given by nature in a natural way, and a person lives, studies, works, makes a career, achieves results, grows, develops. Being an active subject of his own social relations, a person does not refuse to continue the race, no matter what difficulties, misfortunes and sufferings would be experienced. Benathar convinces that all life is continuous suffering, and existence is constant dying. Therefore, it is better not to be born. Materials and Methods. As the main theoretical and methodological direction of research, the dialectical materialist and integrative approaches are used, the realization of which, in conjunction with the synergetic technique, provides a certain result: is convinced that the idea of anti-natalism is inadequate, the idea of giving up life. A systematic approach and a comprehensive assessment of the studied processes provide for the disclosure of the contradictory nature of anti-natalism. Results of the study are presented in the form of conclusions that human life is naturally given by nature itself. Instincts, needs, interests embodied in a person, stimulate to active actions, and he lives. But even if we finish off with all of humanity by agreement, then over time, according to the laws of nature and according to evolutionary theory, man will inevitably, objectively, and naturally reappear. Discussion and Conclusion. The expected effect of the idea of inevitability of rebirth can be the formation of an optimistic orientation of a significant part of the youth, the idea of continuing life and building happiness, development. As a social being, man is universal, and the awareness of this universality allows one to understand one’s purpose – continuous versatile development.

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A note on generalized quasi-contraction

Filomat ◽

10.2298/fil1711091i ◽

2017 ◽

Vol 31 (11) ◽

pp. 3091-3093

Author(s):

Dejan Ilic ◽

Darko Kocev

Keyword(s):

Fixed Point ◽

Fixed Point Theorems ◽

Short Proof

In this paper we give a short proof of the main results of Kumam, Dung and Sitthithakerngkiet (P. Kumam, N.V. Dung, K. Sitthithakerngkiet, A Generalization of Ciric Fixed Point Theorems, FILOMAT 29:7 (2015), 1549-1556).

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Duality Myths

10.1093/oso/9780198809654.003.0009 ◽

2018 ◽

Author(s):

Elizabeth Schechter

Keyword(s):

Narrative Art ◽

Split Brain ◽

Dramatic Form ◽

Natural Way

This chapter addresses the intuitive fascination of the split-brain phenomenon. According to what I call the standard explanation, it is because we ordinarily assume that people are psychologically unified, while split-brain subjects are not psychologically unified, which suggests that we might not be unified either. I offer a different interpretation. One natural way of grappling with people’s failures to conform to various assumptions we make about them is to conceptualize them as having multiple minds. Such multiple-minds models take their most dramatic form in narrative art as duality myths. The split-brain cases grip people in part because the subjects strike them as living embodiments of such myths.

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