scholarly journals New DNA Codes from Cyclic Codes over Mixed Alphabets

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1977
Author(s):  
Hai Q. Dinh ◽  
Sachin Pathak ◽  
Ashish Kumar Upadhyay ◽  
Woraphon Yamaka
Keyword(s):  
Cyclic Codes ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Block Length ◽  
Dna Codes ◽  
Reverse Complement ◽  
Gray Maps ◽  
Cyclic Dna Codes ◽  
One To One ◽  
Necessary And Sufficient

Let R=F4+uF4,withu2=u and S=F4+uF4+vF4,withu2=u,v2=v,uv=vu=0. In this paper, we study F4RS-cyclic codes of block length (α,β,γ) and construct cyclic DNA codes from them. F4RS-cyclic codes can be viewed as S[x]-submodules of Fq[x]⟨xα−1⟩×R[x]⟨xβ−1⟩×S[x]⟨xγ−1⟩. We discuss their generator polynomials as well as the structure of separable codes. Using the structure of separable codes, we study cyclic DNA codes. By using Gray maps ψ1 from R to F42 and ψ2 from S to F43, we give a one-to-one correspondence between DNA codons of the alphabets {A,T,G,C}2,{A,T,G,C}3 and the elements of R,S, respectively. Then we discuss necessary and sufficient conditions of cyclic codes over F4, R, S and F4RS to be reversible and reverse-complement. As applications, we provide examples of new cyclic DNA codes constructed by our results.

DNA cyclic codes over the ring 𝔽2[u,v]/〈u2 − 1,v3 − v,uv − vu〉

2018 ◽  
Vol 11 (03) ◽  
pp. 1850042 ◽  
Author(s):  
Hai Q. Dinh ◽  
Abhay Kumar Singh ◽  
Sukhamoy Pattanayak ◽  
Songsak Sriboonchitta
Keyword(s):  
Cyclic Code ◽  
Cyclic Codes ◽  
Sufficient Conditions ◽  
Gray Map ◽  
Arbitrary Length ◽  
Binary Images ◽  
Dna Codes ◽  
Cyclic Dna Codes ◽  
Living Organisms ◽  
Necessary And Sufficient

In this paper, our main objective is to find out the necessary and sufficient conditions for a cyclic code of arbitrary length over the ring of four elements [Formula: see text] [Formula: see text] to be a reversible cyclic code. We also obtain the structure of cyclic DNA codes of odd length over the ring [Formula: see text], which plays an important role in Computational Biology. Furthermore, we establish a direct link between the elements of ring [Formula: see text] and 64 codons used in the amino acids of living organisms by introducing a Gray map from [Formula: see text] to [Formula: see text]. Among others, binary images of cyclic codes over [Formula: see text] are also investigated. As applications, some cyclic DNA codes over [Formula: see text] using the Gray map are provided.

scholarly journals On cyclic DNA codes over the rings Z4 + wZ4 and Z4 + wZ4 + vZ4 + wvZ4

BIOMATH ◽  
2017 ◽  
Vol 6 (2) ◽  
pp. 1712167 ◽  
Author(s):  
Abdullah Dertli ◽  
Yasemin Cengellenmis
Keyword(s):  
Cyclic Codes ◽  
Finite Ring ◽  
Finite Rings ◽  
Binary Images ◽  
Dna Codes ◽  
Reverse Complement ◽  
Skew Cyclic Codes ◽  
Cyclic Dna Codes

The structures of cyclic DNA codes of odd length over the finite rings R = Z4 + wZ4, w^2 = 2 and S = Z4 + wZ4 + vZ4 + wvZ4; w^2 = 2; v^2 =v; wv = vw are studied. The links between the elements of the rings R, S and 16 and 256 codons are established, respectively. The cyclic codes of odd length over the finite ring R satisfy reverse complement constraint and the cyclic codes of odd length over the finite ring S satisfy reverse constraint and reverse complement constraint are studied. The binary images of the cyclic DNA codes over the finite rings R and S are determined. Moreover, a family of DNA skew cyclic codes over R is constructed, its property of being reverse complement is studied.

scholarly journals Permutation binomials

1990 ◽  
Vol 13 (2) ◽  
pp. 337-342 ◽  
Author(s):  
Charles Small
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Permutation Polynomial ◽  
One To One ◽  
Necessary And Sufficient

A polynomialfover a finite feldFis a permutation polynomial if the mappingF→Fdefined byfis one-to-one. We are concerned here with binomials, that is, polynomials of the shapef=aXi+bXj+c,i>j≥1. Even in this restricted setting, it is impossible to give general necessary and sufficient conditions ona,b,cforfto be a permutation polynomial. We review, and systematize, what is known.

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.

Rearrangeability of Multicast Banyan-Type Networks with Crosstalk Constraints

2017 ◽  
Vol 17 (03n04) ◽  
pp. 1741002
Author(s):  
LI-DA TONG ◽  
YUN-RUEI LI
Keyword(s):  
Parallel Computing ◽  
Communication Networks ◽  
Nonnegative Integer ◽  
Interconnection Networks ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Study Results ◽  
One To One ◽  
Necessary And Sufficient ◽  
Special Case

Multicast interconnection networks are highly demanded in parallel computing and communication networks. In the paper, we give the necessary and sufficient conditions the rearrangeability of the numbers of planes needed for 2j-cast Banyan-type networks with crosstalk constraints and j being a nonnegative integer. Our results of the paper are extended for multicast connection requests, while the previous study results in the paper of Chen et al. are just for a special case for j = 0 and one-to-one connection requests.

Quadruple construction of decomposable double MS-algebras

2020 ◽  
Vol 70 (5) ◽  
pp. 1041-1056
Author(s):  
Abd El-Mohsen Badawy ◽  
Salah El-Din S. Hussein ◽  
Ahmed Gaber
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
One To One ◽  
Necessary And Sufficient

AbstractThis paper is devoted to the study of the class of decomposable double MS-algebras. Necessary and sufficient conditions for a decomposable MS-algebra to be a decomposable double MS-algebra are deduced. We construct decomposable double MS-algebras by means of decomposable MS-quadruples and we prove that there exists a one-to-one correspondence between decomposable double MS-algebras and decomposable MS-quadruples. Moreover, a construction of decomposable K2-algebras (Stone algebras) by means of K2-quadruples (Stone quadruples) is given. We conclude by introducing and characterizing isomorphisms of decomposable double MS-algebras in terms of decomposable MS-quadruples.

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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