THE CORRELATION OF SUBSTITUTION EFFECTS ACROSS POPULATIONS AND GENERATIONS IN THE PRESENCE OF NON-ADDITIVE FUNCTIONAL GENE ACTION

Allele substitution effects at quantitative trait loci (QTL) are part of the basis of quantitative genetics theory and applications such as association analysis and genomic prediction. In the presence of non-additive functional gene action, substitution effects are not constant across populations. We develop an original approach to model the difference in substitution effects across populations as a first order Taylor series expansion from a “focal” population. This expansion involves the difference in allele frequencies and second-order statistical effects (additive by additive and dominance). The change in allele frequencies is a function of relationships (or genetic distances) across populations. As a result, it is possible to estimate the correlation of substitution effects across two populations using three elements: magnitudes of additive, dominance and additive by additive variances; relationships (Nei’s minimum distances or Fst indexes); and assumed heterozygosities. Similarly, the theory applies as well to distinct generations in a population, in which case the distance across generations is a function of increase of inbreeding. Simulation results confirmed our derivations. Slight biases were observed, depending on the non-additive mechanism and the reference allele. Our derivations are useful to understand and forecast the possibility of prediction across populations and the similarity of GWAS effects.

Stability of Bi-Additive Mappings and Bi-Jensen Mappings

Symmetry is repetitive self-similarity. We proved the stability problem by replicating the well-known Cauchy equation and the well-known Jensen equation into two variables. In this paper, we proved the Hyers-Ulam stability of the bi-additive functional equation f(x+y,z+w)=f(x,z)+f(y,w) and the bi-Jensen functional equation 4fx+y2,z+w2=f(x,z)+f(x,w)+f(y,z)+f(y,w).

Stability of a ramanujan type additive functional equation

In this paper, the authors achieve the generalized Ulam - Hyers stability of a Ramanujan Type Additive Functional Equation in Paranormed Spaces and Modular spaces via classical Hyers Method.

Generalized functional inequalities in Banach spaces

In this paper, we solve and investigate the generalized additive functional inequalities ‖ F ( ∑ i = 1 n x i ) - ∑ i = 1 n F ( x i ) ‖ ≤ ‖ F ( 1 n ∑ i = 1 n x i ) - 1 n ∑ i = 1 n F ( x i ) ‖ \left\| {F\left( {\sum\limits_{i = 1}^n {{x_i}} } \right) - \sum\limits_{i = 1}^n {F\left( {{x_i}} \right)} } \right\| \le \left\| {F\left( {{1 \over n}\sum\limits_{i = 1}^n {{x_i}} } \right) - {1 \over n}\sum\limits_{i = 1}^n {F\left( {{x_i}} \right)} } \right\| and ‖ F ( 1 n ∑ i = 1 n x i ) - 1 n ∑ i = 1 n F ( x i ) ‖ ≤ ‖ F ( ∑ i = 1 n x i ) - ∑ i = 1 n F ( x i ) ‖ . \left\| {F\left( {{1 \over n}\sum\limits_{i = 1}^n {{x_i}} } \right) - {1 \over n}\sum\limits_{i = 1}^n {F\left( {{x_i}} \right)} } \right\| \le \left\| {F\left( {\sum\limits_{i = 1}^n {{x_i}} } \right) - \sum\limits_{i = 1}^n {F\left( {{x_i}} \right)} } \right\|. Using the direct method, we prove the Hyers-Ulam stability of the functional inequalities (0.1) in Banach spaces and (0.2) in non-Archimedian Banach spaces.

Hyers–Ulam Stability of Additive Functional Equation Using Direct and Fixed-Point Methods

In this present work, we obtain the solution of the generalized additive functional equation and also establish Hyers–Ulam stability results by using alternative fixed point for a generalized additive functional equation χ ∑ g = 1 l v g = ∑ 1 ≤ g < h < i ≤ l χ v g + v h + v i − ∑ 1 ≤ g < h ≤ l χ v g + v h − l 2 − 5 l + 2 / 2 ∑ g = 1 l χ v g − χ − v g / 2 . where l is a nonnegative integer with ℕ − 0,1,2,3,4 in Banach spaces.

The Correlation Of Substitution Effects Across Populations And Generations In The Presence Of Non-Additive Functional Gene Action

ABSTRACTAllele substitution effects at quantitative trait loci (QTL) are part of the basis of quantitative genetics theory and applications such as association analysis and genomic prediction. In presence of non-additive functional gene action, substitution effects are not constant across populations. We develop an original approach to model the difference in substitution effects across populations as first order Taylor series expansion from a “focal” population. This expansion involves the difference in allele frequencies and second-order statistical effects (additive by additive and dominance). The change in allele frequencies is a function of relationships (or genetic distances) across populations. As a result, it is possible to estimate the correlation of substitution effects across two populations using three elements: magnitudes of additive, dominance and additive by additive variance; relationships across populations (similar to Fst indexes); and functions of heterozygosities at the markers. Similarly, the theory applies as well to distinct generations in a population, in which case the distance across generations is a function of increase of inbreeding. Using published estimates of the needed parameters, we estimate the correlation between substitution effects to be around 0.60 for distinct breeds and higher than 0.9 for generations closer than 5. Simulation results confirmed our derivations although our estimators tended to underestimate the correlation. Our derivations are useful to understand the difficulty in predicting across populations, to forecast the possibility of prediction across populations, and we suggest that they can be useful to disentangle genotype by environment interaction from genotype by genotype interaction.