A multiple marker analysis approach in the framework of the mixed-effects model was developed, allowing all markers of the entire genome to be included simultaneously in the analysis. The approach was extended to multitrait situations. The proposed method is a one-stage process, which simultaneously models the residuals and genetic effects. In addition, it can easily accommodate co-variates, extra sources of variation, fixed or random including polygenic effects and it can easily be generalized to experimental and crossing designs commonly used. The developed approach considered an unstructured co-variance model for the traits residuals and fitted a MULTIPLICATIVE model for the trait by marker effects. The particular MULTIPLICATIVE model considered herein was the factor analytic model. This provided a parsimonious model specification to limit the numberof parameters to be estimated. It was shown through the simulation study that modelling multiple phenotypes in a single linkage analysis simultaneously could markedly increase the power, compared with modelling of each phenotype separately. Correlations among phenotypes can arise from several different causal processes, which may have different implications for the power and performance of the multivariate linkage analysis. Obviously, further studies using the approach suggested herein for multitrait quantitative trait loci (QTL) MAPPING that specifically consider different situations, should be undertaken. Furthermore, the efficiency of the model to distinguish between a pleiotropic QTL and closely linked QTL affecting different traits is another area that needs more investigation.

Our main purpose of this paper is to study the relationship between MULTIPLICATIVE maps and almost MULTIPLICATIVE maps between probabilistic normed algebras. We first derive some properties of invertible elements and their relation with MULTIPLICATIVE maps. Then we show that every complex homomorphism on elements whose probabilistic norm is equal to 1, is bounded. In the following, we give an open problem about the functionally continuous of unital commutative probabilistic Banach algebra. Finally, we prove that every almost MULTIPLICATIVE map that is not a MULTIPLICATIVE map is continuous.

Martindale proved that under some conditions every MULTIPLICATIVE isomorphism between two rings is additive. In this paper, we extend this theorem to a larger class of MAPPINGs and conclude that every MULTIPLICATIVE ( ;  )􀀀 derivation is additive.

In a MULTIPLICATIVE hyperring, the multiplication is a hyperoperation, while the addition is a binary operation. In this paper, the notion of derivation on MULTIPLICATIVE hyperrings is introduced and some related properties are investigated.

A simple connected graph G of order n  3 is a strongly 2-MULTIPLICATIVE if there is an injective MAPPING f: V (G)! f1; 2; : : :; ng such that the induced MAPPING h: A! Z + de ned by h(P) = Q3 i=1 f(vj i ), where j1; j2; j3 2 f1; 2; : : :; ng, and P is the path homotopy class of paths having the vertex set fvj1; vj2; vj3 g, is injective. Let  (n) be the number of distinct path homotopy classes in a strongly 2-MULTIPLICATIVE graph of order n. In this paper we obtain an upper bound and also a lower bound for  (n). Also we prove that triangular ladder, P2 J Cn, Pm J Pn, the graph obtained by duplication of an arbitrary edge by a new vertex in path Pn and the graph obtained by duplicating all vertices by new edges in a path Pn are strongly 2-MULTIPLICATIVE.

In this paper we will give the definition of almost MULTIPLICATIVE linear functionals on commutative Banach algebras and investigate the closeness of these linear functionals with MULTIPLICATIVE linear functionals (AMNM property). َSeveral Banach algebra has this property, but not all of them. Due to relationship between the spectrum of an element of a complex Banach algebra and MULTIPLICATIVE linear functionals. Thus this investigation about almost MULTIPLICATIVE linear functionals give us some information about the spectrum of an element of a Banach algebra. In this paper We will give a very helpful theorem, for this investigation. Also we will show that, if in two commutative Banach algebras A and B, near any almost MULTIPLICATIVE linear functional, there is a MULTIPLICATIVE linear functional, then A×B also has this property and vice versa. We investigate this property for finite product of commutative Banach algebras. Also we will refer to some of the sequential spaces.