Mapping of additive and epistatic QTLs linked to seed longevity in bread wheat (Triticum aestivum L.)

Plant genetic resources are stored and regenerated in > 1750 gene banks storing > 7,000,000 accessions. Since seeds are the primary storage units, research on seed longevity is of particular importance. Quantitative trait loci (QTL) analysis of 15 traits related to seed longevity and dormancy using 7584 high-quality SNPs recorded across 2 years and originated from five production years revealed a total of 46 additive QTLs. Exploration of the QTLs with epistatic effect resulted in the detection of 29 pairs of epistatic QTLs. To our information, this is only the second report of epistatic QTLs for seed longevity in bread wheat. We conclude that in addition to dense genetic maps, the epistatic interaction between loci should be considered to capture more variation which remained unnoticed in additive mapping.

Nearly General Septic Functional Equation

If a mapping can be expressed by sum of a septic mapping, a sextic mapping, a quintic mapping, a quartic mapping, a cubic mapping, a quadratic mapping, an additive mapping, and a constant mapping, we say that it is a general septic mapping. A functional equation is said to be a general septic functional equation provided that each solution of that equation is a general septic mapping. In fact, there are a lot of ways to show the stability of functional equations, but by using the method of G a ˘ vruta, we examine the stability of general septic functional equation ∑ i = 0 8 C 8 i − 1 8 − i f x + i − 4 y = 0 which considered. The method of G a ˘ vruta as just mentioned was given in the reference Gavruta (1994).

Some results on ∗-differential identities in prime rings

Let [Formula: see text] be a prime ring with involution ∗ of the second kind. An additive mapping [Formula: see text] is called generalized derivation if there exists a unique derivation [Formula: see text] such that [Formula: see text] for all [Formula: see text] In this paper, we investigate the structure of [Formula: see text] and describe the possible forms of generalized derivations of [Formula: see text] that satisfy specific ∗-differential identities. Precisely, we study the following situations: (i) [Formula: see text] (ii) [Formula: see text] (iii) [Formula: see text] (iv) [Formula: see text] for all [Formula: see text] Moreover, we construct some examples showing that the restrictions imposed in the hypotheses of our theorems are not redundant.

On Commutativity of Prime Rings with Symmetric Left θ-3- Centralizers

Let R be an associative ring with center Z(R) , I be a nonzero ideal of R and be an automorphism of R . An 3-additive mapping M:RxRxR R is called a symmetric left -3-centralizer if M(u1y,u2 ,u3)=M(u1,u2,u3)(y) holds for all y, u1, u2, u3 R . In this paper , we shall investigate the commutativity of prime rings admitting symmetric left -3-centralizer satisfying any one of the following conditions : (i)M([u ,y], u2, u3) [(u), (y)] = 0 (ii)M((u ∘ y), u2, u3) ((u) ∘ (y)) = 0 (iii)M(u2, u2, u3) (u2) = 0 (iv) M(uy, u2, u3) (uy) = 0 (v) M(uy, u2, u3) (uy) For all u2,u3 R and u ,y I

Asymptotic Stability of the Pexider–Cauchy Functional Equation in Non-Archimedean Spaces

In this paper, we investigated the asymptotic stability behaviour of the Pexider–Cauchy functional equation in non-Archimedean spaces. We also showed that, under some conditions, if ∥f(x+y)−g(x)−h(y)∥⩽ε, then f,g and h can be approximated by additive mapping in non-Archimedean normed spaces. Finally, we deal with a functional inequality and its asymptotic behaviour.

On Centralizers of 2-torsion Free Semiprime Gamma Rings

In this paper, we prove that; Let M be a 2-torsion free semiprime which satisfies the condition for all and α, β . Consider that as an additive mapping such that holds for all and α , then T is a left and right centralizer.

A result related to derivations on unital semiprime rings

The purpose of this paper is to prove the following result. Let n≥3 be some fixed integer and let R be a (n+1)!2n-2-torsion free semiprime unital ring. Suppose there exists an additive mapping D: R→ R satisfying the relation for all x ∈ R. In this case D is a derivation. The history of this result goes back to a classical result of Herstein, which states that any Jordan derivation on a 2-torsion free prime ring is a derivation.

Полудифференцирования в первичных кольцах

Let $\mathscr{R}$ be a prime ring with the extended centroid $\mathscr{C}$ and the Matrindale quotient ring $\mathscr{Q}$. An additive mapping $\mathscr{F}:\mathscr{R}\rightarrow \mathscr{R}$ is called a semiderivation associated with a mapping $\mathscr{G}: \mathscr{R}\rightarrow \mathscr{R}$, whenever $ \mathscr{F}(xy)=\mathscr{F}(x)\mathscr{G}(y)+x\mathscr{F}(y)= \mathscr{F}(x)y+ \mathscr{G}(x)\mathscr{F}(y) $ and $ \mathscr{F}(\mathscr{G}(x))= \mathscr{G}(\mathscr{F}(x))$ holds for all $x, y \in \mathscr{R}$. In this manuscript, we investigate and describe the structure of a prime ring $\mathscr{R}$ which satisfies $\mathscr{F}(x^m\circ y^n)\in \mathscr{Z(R)}$ for all $x, y \in \mathscr{R}$, where $m,n \in \mathbb{Z}^+$ and $\mathscr{F}:\mathscr{R}\rightarrow \mathscr{R}$ is a semiderivation with an~automorphism $\xi$ of $\mathscr{R}$. Further, as an application of our ring theoretic results, we discussed the nature of $\mathscr{C}^*$-algebras. To be more specific, we obtain for any primitive $\mathscr{C}^*$-algebra $\mathscr{A}$. If an anti-automorphism $ \zeta: \mathscr{A} \to \mathscr{A}$ satisfies the relation $(x^n)^\zeta+x^{n*}\in \mathscr{Z}(\mathscr{A})$ for every ${x,y}\in \mathscr{A},$ then $\mathscr{A}$ is $\mathscr{C}^{*}-\mathscr{W}_{4}$-algebra, i.\,e., $\mathscr{A}$ satisfies the standard identity $\mathscr{W}_4(a_1,a_2,a_3,a_4)=0$ for all $a_1,a_2,a_3,a_4\in \mathscr{A}$.

Symmetric Reverse Gamma *-4-Centralizers on Semiprime Gamma Rings with Involution

Let M be a -ring with involution . In this paper , we will introduce the concept of symmetric left(right) reverse *-4-centralizer of M . Then, we proved that the 4-additive mapping T:MxMxMxMM is a reverse *-4-centralizer of M under certain conditions .

Identities related to generalized derivations and jordan (*,*)-derivations

The main purpose of this research is to characterize generalized (left) derivations and Jordan (*,*)-derivations on Banach algebras and rings using some functional identities. Let A be a unital semiprime Banach algebra and let F,G : A ? A be linear mappings satisfying F(x) =-x2G(x-1) for all x ? Inv(A), where Inv(A) denotes the set of all invertible elements of A. Then both F and G are generalized derivations on A. Another result in this regard is as follows. Let A be a unital semiprime algebra and let n > 1 be an integer. Let f,g : A ? A be linear mappings satisfying f (an) = nan-1g(a) = ng(a)an-1 for all a ? A. If g(e) ? Z(A), then f and g are generalized derivations associated with the same derivation on A. In particular, if A is a unital semisimple Banach algebra, then both f and 1 are continuous linear mappings. Moreover, we define a (*,*)-ring and a Jordan (*,*)-derivation. A characterization of Jordan (*,*)-derivations is presented as follows. Let R be an n!-torsion free (*,*)-ring, let n > 1 be an integer and let d : R ? R be an additive mapping satisfying d(an) = ?nj =1 a*n-jd(a)a* j-1 for all a ? R. Then d is a Jordan (*,*)-derivation on R. Some other functional identities are also investigated.