scholarly journals Determining When an Algebra Is an Evolution Algebra

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1349 ◽  
Author(s):  
Miguel D. Bustamante ◽  
Pauline Mellon ◽  
M. Victoria Velasco
Keyword(s):  
Asexual Reproduction ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Associative Algebras ◽  
Dimensional Algebra ◽  
Small Perturbations ◽  
Classical Genetic ◽  
Evolution Algebra ◽  
Necessary And Sufficient

Evolution algebras are non-associative algebras that describe non-Mendelian hereditary processes and have connections with many other areas. In this paper, we obtain necessary and sufficient conditions for a given algebra A to be an evolution algebra. We prove that the problem is equivalent to the so-called SDC problem, that is, the simultaneous diagonalisation via congruence of a given set of matrices. More precisely we show that an n-dimensional algebra A is an evolution algebra if and only if a certain set of n symmetric n×n matrices {M1,…,Mn} describing the product of A are SDC. We apply this characterisation to show that while certain classical genetic algebras (representing Mendelian and auto-tetraploid inheritance) are not themselves evolution algebras, arbitrarily small perturbations of these are evolution algebras. This is intringuing, as evolution algebras model asexual reproduction, unlike the classical ones.

Identities of Non-Associative Algebras

1965 ◽  
Vol 17 ◽  
pp. 78-92 ◽  
Author(s):  
J. Marshall Osborn
Keyword(s):  
Sufficient Conditions ◽  
Interesting Property ◽  
Necessary And Sufficient Conditions ◽  
Partial Ordering ◽  
Associative Algebras ◽  
Commutative Algebras ◽  
Odd Degree ◽  
Necessary And Sufficient

In the first part of this paper we define a partial ordering on the set of all homogeneous identities and find necessary and sufficient conditions that an identity does not imply any identity lower than it in the partial ordering (we call such an identity irreducible). Perhaps the most interesting property established for irreducible identities is that they are skew-symmetric in any two variables of the same odd degree and symmetric in any two variables of the same even degree. The results of the first section are applied to commutative algebras in the remainder of the paper.

scholarly journals On the Classification of Bol-Moufang Type of Some Varieties of Quasi Neutrosophic Triplet Loop (Fenyves BCI-Algebras)

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 427 ◽  
Author(s):  
Tèmítọ́pẹ́ Jaíyéọlá ◽  
Emmanuel Ilojide ◽  
Memudu Olatinwo ◽  
Florentin Smarandache
Keyword(s):  
Research Finding ◽  
Research Work ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Algebraic Structures ◽  
Associative Algebras ◽  
Necessary And Sufficient

In this paper, Bol-Moufang types of a particular quasi neutrosophic triplet loop (BCI-algebra), chritened Fenyves BCI-algebras are introduced and studied. 60 Fenyves BCI-algebras are introduced and classified. Amongst these 60 classes of algebras, 46 are found to be associative and 14 are found to be non-associative. The 46 associative algebras are shown to be Boolean groups. Moreover, necessary and sufficient conditions for 13 non-associative algebras to be associative are also obtained: p-semisimplicity is found to be necessary and sufficient for a F 3 , F 5 , F 42 and F 55 algebras to be associative while quasi-associativity is found to be necessary and sufficient for F 19 , F 52 , F 56 and F 59 algebras to be associative. Two pairs of the 14 non-associative algebras are found to be equivalent to associativity ( F 52 and F 55 , and F 55 and F 59 ). Every BCI-algebra is naturally an F 54 BCI-algebra. The work is concluded with recommendations based on comparison between the behaviour of identities of Bol-Moufang (Fenyves’ identities) in quasigroups and loops and their behaviour in BCI-algebra. It is concluded that results of this work are an initiation into the study of the classification of finite Fenyves’ quasi neutrosophic triplet loops (FQNTLs) just like various types of finite loops have been classified. This research work has opened a new area of research finding in BCI-algebras, vis-a-vis the emergence of 540 varieties of Bol-Moufang type quasi neutrosophic triplet loops. A ‘Cycle of Algebraic Structures’ which portrays this fact is provided.

(𝜃,ω)-Twisted Radford’s Hom-biproduct and ϖ-Yetter–Drinfeld modules for Hom-Hopf algebras

2020 ◽  
Vol 19 (03) ◽  
pp. 2050046
Author(s):  
Xiao-Li Fang ◽  
Tae-Hwa Kim
Keyword(s):  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Tensor Products ◽  
Product Formula ◽  
Necessary And Sufficient Conditions ◽  
Drinfeld Modules ◽  
Associative Algebras ◽  
Necessary And Sufficient

To unify different definitions of smash Hom-products in a Hom-bialgebra [Formula: see text], we firstly introduce the notion of [Formula: see text]-twisted smash Hom-product [Formula: see text]. Secondly, we find necessary and sufficient conditions for the twisted smash Hom-product [Formula: see text] and the twisted smash Hom-coproduct [Formula: see text] to afford a Hom-bialgebra, which generalize the well-known Radford’s biproduct and the Hom-biproduct obtained in [H. Li and T. Ma, A construction of the Hom-Yetter–Drinfeld category, Colloq. Math. 137 (2014) 43–65]. Furthermore, we introduce the notion of the category of [Formula: see text]-Yetter-Drinfeld modules which unifies the ones of Hom-Yetter Drinfeld category appeared in [H. Li and T. Ma, A construction of the Hom-Yetter–Drinfeld category, Colloq. Math. 137 (2014) 43–65] and [A. Makhlouf and F. Panaite, Twisting operators, twisted tensor products and smash products for Hom-associative algebras, J. Math. Glasgow 513–538 (2016) 58]. Finally, we prove that the [Formula: see text]-twisted Radford’s Hom-biproduct [Formula: see text] is a Hom-bialgebra if and only if [Formula: see text] is a Hom-bialgebra in the category of [Formula: see text]-Yetter–Drinfeld modules [Formula: see text], generalizing the well-known Majid’s conclusion.

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.

On the domain of Riesz mean in the space Ls

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 925-940 ◽  
Author(s):  
Medine Yeşilkayagil ◽  
Feyzi Başar
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Double Sequence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Double Sequences ◽  
Riesz Mean ◽  
Double Sequence Space ◽  
Necessary And Sufficient ◽  
Barrelled Space

Let 0 < s < ?. In this study, we introduce the double sequence space Rqt(Ls) as the domain of four dimensional Riesz mean Rqt in the space Ls of absolutely s-summable double sequences. Furthermore, we show that Rqt(Ls) is a Banach space and a barrelled space for 1 ? s < 1 and is not a barrelled space for 0 < s < 1. We determine the ?- and ?(?)-duals of the space Ls for 0 < s ? 1 and ?(bp)-dual of the space Rqt(Ls) for 1 < s < 1, where ? ? {p, bp, r}. Finally, we characterize the classes (Ls:Mu), (Ls:Cbp), (Rqt(Ls) : Mu) and (Rqt(Ls):Cbp) of four dimensional matrices in the cases both 0 < s < 1 and 1 ? s < 1 together with corollaries some of them give the necessary and sufficient conditions on a four dimensional matrix in order to transform a Riesz double sequence space into another Riesz double sequence space.

Sign in / Sign up

Export Citation Format

Share Document