Generation of the alternating group by modular additions

2019 ◽  
Vol 29 (5) ◽  
pp. 303-309
Author(s):  
Fedor M. Malyshev
Keyword(s):  
Permutation Group ◽  
Cartesian Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Local Action ◽  
Alternating Group ◽  
Permutation Groups ◽  
Cartesian Products ◽  
Strong Connectedness ◽  
Necessary And Sufficient

Abstract The paper is concerned with systems of generators of permutation groups on Cartesian products of residue rings. Each separate permutation from the system of generators is constructed on the basis of additions, is characterized by the local action, and leaves fixed the major parts of the components of the element being transformed. A criterion of 2-transitivity of the generated permutation group is given in the form of the strong connectedness of the digraph which corresponds to the system of generators and which is defined on the set of numbers of residue rings in the Cartesian product. Necessary and sufficient conditions under which this group contains an alternating group are formulated.

scholarly journals On Cartesian products with small crossing numbers

2012 ◽  
Vol 28 (1) ◽  
pp. 67-75
Author(s):  
MARIAN KLESC ◽  
◽  
JANA PETRILLOVA ◽  
Keyword(s):  
Cartesian Product ◽  
Sufficient Conditions ◽  
Crossing Number ◽  
Necessary And Sufficient Conditions ◽  
Line Graphs ◽  
Cartesian Products ◽  
Crossing Numbers ◽  
Necessary And Sufficient

Kulli at al. started to characterize line graphs with crossing number one. In this paper, the similar problems were solved for the Cartesian products of two graphs. The necessary and sufficient conditions are given for all pairs of graphs G1 and G2 for which the crossing number of their Cartesian product G1 × G2 is one or two.

scholarly journals Group closures of one-to-one transformations

2001 ◽  
Vol 64 (2) ◽  
pp. 177-188 ◽  
Author(s):  
Inessa Levi
Keyword(s):  
Normal Subgroup ◽  
Symmetric Group ◽  
Permutation Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Alternating Group ◽  
One To One ◽  
Necessary And Sufficient ◽  
Infinite Set

For a semigroup S of transformations of an infinite set X let Gs be the group of all the permutations of X that preserve S under conjugation. Fix a permutation group H on X and a transformation f of X, and let 〈f: H〉 = 〈{hfh−1: h ∈ H}〉 be the H-closure of f. We find necessary and sufficient conditions on a one-to-one transformation f and a normal subgroup H of the symmetric group on X to satisfy G〈f:H〉 = H. We also show that if S is a semigroup of one-to-one transformations of X and GS contains the alternating group on X then Aut(S) = Inn(S) ≅ GS.

scholarly journals Endomorphism regular Ockham algebras of finite Boolean type

1997 ◽  
Vol 39 (1) ◽  
pp. 99-110 ◽  
Author(s):  
T. S. Blyth ◽  
H. J. Silva
Keyword(s):  
Inverse Semigroup ◽  
Dual Space ◽  
Cartesian Product ◽  
Sufficient Conditions ◽  
Boolean Lattice ◽  
Necessary And Sufficient Conditions ◽  
Ockham Algebra ◽  
Cyclic Groups ◽  
Ockham Algebras ◽  
Necessary And Sufficient

AbstractIf (L; ƒ) is an Ockham algebra with dual space (X; g), then it is known that the semigroup of Ockham endomorphisms on L is (anti-)isomorphic to the semigroup Λ(X; g) of continuous order-preserving mappings on X that commute with g. Here we consider the case where L is a finite boolean lattice and ƒ is a bijection. We begin by determining the size of Λ(X;g), and obtain necessary and sufficient conditions for this semigroup to be regular or orthodox. We also describe its structure when it is a group, or an inverse semigroup that is not a group. In the former case it is a cartesian product of cyclic groups and in the latter a cartesian product of cyclic groups each with a zero adjoined.

scholarly journals Metric products and continuation of isotone functions

2014 ◽  
Vol 64 (1) ◽  
Author(s):  
O. Dovgoshey ◽  
E. Petrov ◽  
G. Kozub
Keyword(s):  
Metric Spaces ◽  
Cartesian Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Moduli Of Continuity ◽  
Necessary And Sufficient

AbstractLet ℝ+ = [0,∞) and let A ⊆ ℝ+n. We have found the necessary and sufficient conditions under which a function Φ: A → ℝ+ has an isotone subadditive continuation on ℝ+n. It allows us to describe the metrics, defined on the Cartesian product X 1×...×X n of given metric spaces $$\left( {X_1 ,d_{X_1 } } \right), \ldots ,\left( {X_n ,d_{X_n } } \right)$$, generated by the isotone metric preserving functions on ℝ+n. It is also shown that the isotone metric preserving functions Φ: ℝ+n → ℝ+ coincide with the first moduli of continuity of the nonconstant bornologous functions g: ℝ+n → ℝ+.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

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