Monge-Ampere equations and generalized complex geometry. The two-dimensional case

We associate an integrable generalized complex structure to each 2-dimensional symplectic Monge-Amp\`ere equation of divergent type and, using the Gualtieri $\bar{\partial}$ operator, we characterize the conservation laws and the generating function of such equation as generalized holomorphic objects

Complex solutions of Monge-Amp\`ere equations

We describe a method to reduce partial differential equations of Monge-Amp\`ere type in 4 variables to complex partial differential equations in 2 variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge-Amp\`ere equations and the Plebanski equations.Comment: 16 pages, 5 tables To appear in Journal of Geometry and Physic

Genetic Relationship Between the United States and Canadian Holstein Bull Populations

The average additive genetic relationship and degree of connectedness between American and Canadian Holstein AI bull populations were estimated. This project was undertaken to determine the feasibility of a joint United States-Canadian sire evaluation to estimate genetic base differences between the two countries‘Holstein bull populations. Data were provided by USDA and Agriculture Canada for bulls evaluated in each country. Bulls were designated as American, Canadian, or dual national origin based on their country of registration and national origin of their parents. A total of 13,079 American, 1683 Canadian, and 256 dual origin bulls were included in the inverse relationship matrix. When both sire and maternal grandsire relationships were included in the matrix, there were 174 disconnected groups; however, 99% of the American bulls and 97% of the Canadian bulls were in a single group. The average aij between the American and Canadian population was 4.6×10−5. Despite the low average additive relationship between the two national populations, the high degree of connectedness in the inverse relationship matrix, when using both sire and maternal grandsire relationships, suggests sufficient genetic ties between the two populations to conduct a meaningful joint sire evaluation

Implications of avoiding overlap between training and testing data sets when evaluating genomic predictions of genetic merit

The aim of this study was to evaluate and quantify the importance of avoiding overlap between training and testing subsets of data when evaluating the effectiveness of predictions of genetic merit based on genetic markers. Genomic selection holds great potential for increasing the accuracy of selection in young bulls and is likely to lead quickly to more widespread use of these young bulls with a shorter generation interval and faster genetic improvement. Practical implementations of genomic selection in dairy cattle commonly involve results of national genetic evaluations being used as the dependent variable to evaluate the predictive ability of genetic markers. Selection index theory was used to demonstrate how ignoring correlations among errors of prediction between animals in training and testing sets could result in overestimates of accuracy of genomic predictions. Correlations among errors of prediction occur when estimates of genetic merit of training animals used in prediction are taken from the same genetic evaluation as estimates for validation of animals. Selection index theory was used to show a substantial degree of error correlation when animals used for testing genomic predictions are progeny of training animals, when heritability is low, and when the number of recorded progeny for both training and testing animals is low. Even when training involves a dependent variable that is not influenced by the progeny records of testing animals (i.e., historic proofs), error correlations can still result from records of relatives of training animals contributing to both the historic proofs and the predictions of genetic merit of testing animals. A simple simulation was used to show how an error correlation could result in spurious confirmation of predictive ability that was overestimated in the training population because of ascertainment bias. Development of a method of testing genomic selection predictions that allows unbiased testing when training and testing variables are estimated breeding values from the same genetic evaluation would simplify training and testing of genomic predictions. In the meantime, a 4-step approach for separating records used for training from those used for testing after correction of fixed effects is suggested when use of progeny averages of adjusted records (e.g., daughter yield deviations) would result in inefficient use of the information available in the data.</p

Short communication: Characterization of the genome-wide linkage disequilibrium in 2 divergent selection lines of dairy cows

The objective of this study was to describe results of a genome-wide map of single nucleotide polymorphisms (SNP) and assess the linkage disequilibrium (LD) level in 2 divergent selection lines of dairy cows. DNA extracted from 299 Holstein cows was used to determine genotypes in 54,001 SNP loci using the BovineSNP50 array (Illumina Inc., San Diego, CA). Animals were from 2 genetic lines (166 genetically selected for fat and protein yield vs. 133 controls) raised on an experimental farm. Data edits removed loci with a major allele frequency greater than 0.95, genotypes in fewer than 100 cows, and missing valid chromosomal assignment or position. After edits, 41,859 loci (77.5% of the original total) were kept for further analysis. Linkage disequilibrium (LD) values were calculated for all possible syntenic SNP locus pairs located within intervals of 1 million base pairs, as the squared correlation between alleles. Pairwise haplotypes were determined using parsimony. Linkage disequilibrium was calculated for all animals and then for each genetic line separately. The average LD calculated across all chromosomes was 0.069, 0.071, and 0.075 for all, control, and select line cows, respectively. Genetic line had a statistically significant effect on LD. Of all locus pairs studied, 53,487 to 95,279 (depending on the data set) were in LD &gt; 0.30, which may be considered the minimum useful for mapping purposes and genomic selection. Useful LD was mostly found between adjacent pairs located within 30,000 to 50,000 bases. A few locus pairs (844-1,070 in the 3 data sets) were found in almost perfect (&gt; 0.99) LD. The overall product-moment correlation of LD values between the control and select lines was 0.79 (significantly different from 1), ranging from 0.71 to 0.84 for different chromosomes. Looking at this correlation by SNP pair distance revealed that persistence of LD phase across the 2 lines extended chiefly for 200,000 bases. Selection is likely to have affected LD in the studied cow population. These results may be useful to gene detection and genome-wide association studies.</p

Genetic profile of total body energy content of Holstein cows in the first three lactations

Weekly total body energy content (TBEC) was calculated for 444 Holstein cows in their first 3 lactations. These calculations were based on body lipid and protein changes predicted from weekly changes in body condition score and live weight of each cow. In first lactation, cows lost TBEC during the initial 8 wk, regained it by wk 22, and continued to build up their reserves until wk 37. Cows started lactations 2 and 3 with considerable reserves from the dry period that they used during the first 13 wk of lactation. Variance components for TBEC were estimated using random regression analysis allowing for heterogeneous residual variance. The genetic variance increased within each lactation, suggesting that the genetic component becomes more important as lactation progresses. The genetic correlations between very early ( wk 1 to 4) and later stages of first lactation were near zero but they increased considerably between later lactation stages. Genetic correlations between TBEC on wk 5 of first lactation and the remainder of this lactation ranged from 0.64 for the more distant weeks to 0.99 for the immediately subsequent weeks. Genetic correlations with TBEC in second lactation were moderately high (0.68 to 0.70) for the early weeks ( 1 to 8) and decreased gradually to 0.56 for weeks at the end of lactation. For third lactation, these estimates ranged from 0.53 to 0.63. Genetic correlation estimates of TBEC in wk 12 of first lactation with subsequent first-lactation weeks varied from 0.79 to 0.99, whereas they ranged from 0.65 to 0.77 and from 0.57 to 0.68 in second and third lactations, respectively. The genetic correlation between TBEC in later weeks of first lactation and the rest of productive life increased as first lactation progressed, but the improvement diminished. Weekly genetic evaluations for first-lactation TBEC were used to predict second- and third-lactation energy content. The accuracy of these predictions increased with progressing weeks in first lactation, but about three-fourths of the improvement occurred by wk 5. Our results suggest that TBEC calculated after a month from the first calving may give useful information about the future energy content of a cow.</p

Evaluation of body condition score measured throughout lactation as an indicator of fertility in dairy cattle

Body condition score (BCS) records of primiparous Holstein cows were analyzed both as a single measure per animal and as repeated measures per sire of cow. The former resulted in a single, average, genetic evaluation for each sire, and the latter resulted in separate genetic evaluations per day of lactation. Repeated measure analysis yielded genetic correlations of less than unity between days of lactation, suggesting that BCS may not be the same trait across lactation. Differences between daily genetic evaluations on d 10 or 30 and subsequent daily evaluations were used to assess BCS change at different stages of lactation. Genetic evaluations for BCS level or change were used to estimate genetic correlations between BCS measures and fertility traits in order to assess the capacity of BCS to predict fertility. Genetic correlation estimates with calving interval and non-return rate were consistently higher for daily BCS than single measure BCS evaluations, but results were not always statistically different. Genetic correlations between BCS change and fertility traits were not significantly different from zero. The product of the accuracy of BCS evaluations with their genetic correlation with the UK fertility index, comprising calving interval and non-return rate, was consistently higher for daily than for single BCS evaluations, by 28 to 53%. This product is associated with the conceptual correlated response in fertility from BCS selection and was highest for early (d 10 to 75) evaluations.</p