scholarly journals On elations in semi-transitive planes

1982 ◽  
Vol 5 (1) ◽  
pp. 159-164
Author(s):  
N. L. Johnson
Keyword(s):  
Translation Plane ◽  
Hall Plane ◽  
Even Order ◽  
Affine Elation

Letπbe a semi-transitive translation plane of even order with reference to the subplaneπ0. Ifπadmits an affine elation fixingπ0for each axis inπ0and the order ofπ0is not2or8, thenπis a Hall plane.

scholarly journals A characterization of generalized Hall planes

1972 ◽  
Vol 6 (1) ◽  
pp. 61-67 ◽  
Author(s):  
N.L. Johnson
Keyword(s):  
Translation Plane ◽  
Hall Plane ◽  
Odd Order ◽  
Class 1 ◽  
Even Order ◽  
Semifield Planes ◽  
Dickson Semifield

We prove that a translation plane π of odd order is a generalized Hall plane if and only if π is derived from a translation plane of semi-translation class 1–3a. Also, a derivable translation plane of even order and class 1–3a derives a generalized Hall plane. We also show that the generalized Hall planes of Kirkpatrick form a subclass of the class of planes derived from the Dickson semifield planes.

scholarly journals The geometry ofGL(2,q)in translation planes of even orderq2

1978 ◽  
Vol 1 (4) ◽  
pp. 447-458 ◽  
Author(s):  
N. L. Johnson
Keyword(s):  
Translation Plane ◽  
Collineation Group ◽  
Translation Planes ◽  
Even Order

In this article we show the following: Letπbe a translation plane of even orderq2that admitsGL(2,q)as a collineation group. Thenπis either Desarguesian, Hall or Ott-Schaeffer.

On Translation Planes which Admit Solvable Autotopism Groups Having a Large Slope Orbit

1984 ◽  
Vol 36 (5) ◽  
pp. 769-782 ◽  
Author(s):  
Vikram Jha
Keyword(s):  
Translation Plane ◽  
Collineation Group ◽  
Affine Plane ◽  
Translation Planes ◽  
Hall Plane ◽  
Autotopism Group ◽  
Affine Translation ◽  
Planar Group ◽  
Affine Translation Plane

Our main object is to prove the following result.THEOREM C. Let A be an affine translation plane of order qr ≧ q2 suchthatl∞, the line at infinity, coincides with the translation axis of A. Suppose G is a solvable autotopism group of A that leaves invariant a set Δ of q + 1 slopes and acts transitively on l∞ \ Δ.Then the order of A is q2.An autotopism group of any affine plane A is a collineation group G that fixes at least two of the affine lines of A; if in fact the fixed elements of G form a subplane of A we call G a planar group. When A in the theorem is a Hall plane [4, p. 187], or a generalized Hall plane ([13]), G can be chosen to be a planar group.

scholarly journals Translation planes of even order in which the dimension has only one odd factor

1980 ◽  
Vol 3 (4) ◽  
pp. 675-694 ◽  
Author(s):  
T. G. Ostrom
Keyword(s):  
Normal Subgroup ◽  
Translation Plane ◽  
Cyclic Subgroup ◽  
Prime Factor ◽  
Translation Planes ◽  
Translation Complement ◽  
Finite Translation ◽  
Finite Translation Plane ◽  
Irreducible Subgroup ◽  
Even Order

LetGbe an irreducible subgroup of the linear translation complement of a finite translation plane of orderqdwhereqis a power of2.GF(q)is in the kernel andd=2srwhereris an odd prime. A prime factor of|G|must divide(qd+1)∏i=1d(qi−1).One possibility (there are no known examples) is thatGhas a normal subgroupWwhich is aW-group for some primeW.The maximal normal subgroup0(G)satisfies one of the following:1. Cyclic. 2. Normal cyclic subgroup of indexrand the nonfixed-point-free elements in0(G)have orderr. 3.0(G)contains a groupWas above.

scholarly journals Existence of Multiple Positive Solutions for Even Order Multi-Point Boundary Value Problems

2007 ◽  
Vol 14 (4) ◽  
pp. 775-792
Author(s):  
Youyu Wang ◽  
Weigao Ge
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Multiple Positive Solutions ◽  
Point Boundary ◽  
Even Order

Abstract In this paper, we consider the existence of multiple positive solutions for the 2𝑛th order 𝑚-point boundary value problem: where (0,1), 0 < ξ 1 < ξ 2 < ⋯ < ξ 𝑚–2 < 1. Using the Leggett–Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The associated Green's function for the above problem is also given.

scholarly journals Even-order differential equation with continuous delay: nonexistence criteria of Kneser solutions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ali Muhib ◽  
M. Motawi Khashan ◽  
Osama Moaaz
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Delay Argument ◽  
Even Order ◽  
Continuous Delay ◽  
New Criteria

AbstractIn this paper, we study even-order DEs where we deduce new conditions for nonexistence Kneser solutions for this type of DEs. Based on the nonexistence criteria of Kneser solutions, we establish the criteria for oscillation that take into account the effect of the delay argument, where to our knowledge all the previous results neglected the effect of the delay argument, so our results improve the previous results. The effectiveness of our new criteria is illustrated by examples.

scholarly journals Two-color two-dimensional terahertz spectroscopy: A new approach for exploring even-order nonlinearities in the nonperturbative regime

2021 ◽  
Vol 154 (15) ◽  
pp. 154203
Author(s):  
Michael Woerner ◽  
Ahmed Ghalgaoui ◽  
Klaus Reimann ◽  
Thomas Elsaesser
Keyword(s):  
Terahertz Spectroscopy ◽  
Two Dimensional ◽  
New Approach ◽  
Even Order

A nonlocal problem for a differential operator of even order with involution

2020 ◽  
Vol 26 (2) ◽  
pp. 297-307
Author(s):  
Petro I. Kalenyuk ◽  
Yaroslav O. Baranetskij ◽  
Lubov I. Kolyasa
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Riesz Basis ◽  
Existence And Uniqueness ◽  
Spectral Properties ◽  
Nonlocal Problem ◽  
Even Order

AbstractWe study a nonlocal problem for ordinary differential equations of {2n}-order with involution. Spectral properties of the operator of this problem are analyzed and conditions for the existence and uniqueness of its solution are established. It is also proved that the system of eigenfunctions of the analyzed problem forms a Riesz basis.

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