Legendre Polynomials Of Order Research Articles

Selection for Test-Day Milk Yield and Thermotolerance in Brazilian Holstein Cattle

Simple SummaryInterest in selection for milk yield and thermotolerance in cattle has grown, since heat stress has caused great losses in milk yield. However, few studies on how to carry out concurrent selection are available. Milk yield was analyzed by traditional methods, including heat stress indicators, in genetic evaluation. The results showed that the best sires for milk yield are not the best for heat tolerance, and only a small proportion of individuals have the aptitude for joint selection. Despite a small population fraction allowed for joint selection, sufficient genetic variability for selecting more resilient sires was found, which promoted concomitant genetic gains in milk yield and thermotolerance.Intense selection for milk yield has increased environmental sensitivity in animals, and currently, heat stress is an expensive problem in dairy farming. The objectives were to identify the best model for characterizing environmental sensitivity in Holstein cattle, using the test-day milk yield (TDMY) combined with the temperature–humidity index (THI), and identify sires genetically superior for heat-stress (HS) tolerance and milk yield, through random regression. The data comprised 94,549 TDMYs of 11,294 first-parity Holstein cows in Brazil, collected from 1997 to 2013. The yield data were fitted to Legendre orthogonal polynomials, linear splines and the Wilmink function. The THI (the average of two days before the dairy control) was used as an environmental gradient. An animal model that fitted production using a Legendre polynomials of quartic order for the days in milk and quadratic equations for the THI presented a better quality of fit (Akaike’s information criterion (AIC) and Bayesian information criterion (BIC)). The Spearman correlation coefficient of greatest impact was 0.54, between the top 1% for TDMY and top 1% for HS. Only 9% of the sires showed plasticity and an aptitude for joint selection. Thus, despite the small population fraction allowed for joint selection, sufficient genetic variability for selecting more resilient sires was found, which promoted concomitant genetic gains in milk yield and thermotolerance.

Using Random Regression Models to Genetically Evaluate Functional Longevity Traits in North American Angus Cattle.

Simple SummaryCattle longevity is usually defined as the duration of life of a cow from first calving to death. In addition to a longer lifespan, it is crucial that cows are productive throughout their lives. Incorporating optimal indicators of productive longevity in breeding schemes will directly improve the economic profitability of the beef cattle herd and long-term sustainability of the industry. Thus, the impact of different longevity indicators in the selection of North American Angus cattle was evaluated and optimal parameters were defined to perform the evaluations.This study aimed to propose novel longevity indicators by comparing genetic parameters for traditional (TL; i.e., the cow’s lifespan after the first calving) and functional (FL; i.e., how long the cow stayed in the herd while also calving; assuming no missing (FLa) or missing (FLb) records for unknown calving) longevity, considering different culling reasons (natural death, structural problems, disease, fertility, performance, and miscellaneous). Longevity definitions were evaluated from 2 to 15 years of age, using single- and multiple-trait Bayesian random regression models (RRM). The RRM fitting heterogenous residual variance and fourth order Legendre polynomials were considered as the optimal models for the majority of longevity indicators. The average heritability estimates over ages for FLb (from 0.08 to 0.25) were always higher than those for FLa (from 0.07 to 0.19), and higher or equal to the ones estimated for TL (from 0.07 to 0.23), considering the different culling reasons. The average genetic correlations estimated between ages were low to moderate (~0.40), for all longevity definitions and culling reasons. However, removing the extreme ages (i.e., 2 and >12 years) increased the average correlation between ages (from ~0.40 to >0.70). The genetic correlations estimated between culling reasons were low (0.12 and 0.20 on average, considering all ages and ages between 3 and 12 years old, respectively), indicating that longevity based on different culling reasons should be considered as different traits in the genetic evaluations. Higher average genetic correlations (estimated from 3 to 12 years old) were observed between TL and FLb (0.73) in comparison to TL and FLa (0.64), or FLa and FLb (0.65). Consequently, a higher average proportion of commonly-selected sires, for the top 1% sires, was also observed between TL and FLb (91.74%), compared to TL and FLa (59.68%), or FLa and FLb (61.01%). Higher prediction accuracies for the expected daughter performances (calculated based on the pedigree information) were obtained for FLb in comparison to TL and FLa. Our findings indicate that FLb is preferred for the genetic evaluation of longevity. In addition, it is recommended including multiple longevity traits based on different groups of culling reasons in a selection sub-index, as they are genetically-different traits. Genetic selection based on breeding values at the age of four years is expected to result in greater selection responses for increased longevity in North American Angus cattle.

Multiple-trait, random regression, and compound symmetry models for analyzing multi-environment trials in maize breeding.

An efficient and informative statistical method to analyze genotype-by-environment interaction (GxE) is needed in maize breeding programs. Thus, the objective of this study was to compare the effectiveness of multiple-trait models (MTM), random regression models (RRM), and compound symmetry models (CSM) in the analysis of multi-environment trials (MET) in maize breeding. For this, a data set with 84 maize hybrids evaluated across four environments for the trait grain yield (GY) was used. Variance components were estimated by restricted maximum likelihood (REML), and genetic values were predicted by best linear unbiased prediction (BLUP). The best fit MTM, RRM, and CSM were identified by the Akaike information criterion (AIC), and the significance of the genetic effects were tested using the likelihood ratio test (LRT). Genetic gains were predicted considering four selection intensities (5, 10, 15, and 20 hybrids). The selected MTM, RRM, and CSM models fit heterogeneous residuals. Moreover, for RRM the genetic effects were modeled by Legendre polynomials of order two. Genetic variability between maize hybrids were assessed for GY. In general, estimates of broad-sense heritability, selective accuracy, and predicted selection gains were slightly higher when obtained using MTM and RRM. Thus, considering the criterion of parsimony and the possibility of predicting genetic values of hybrids for untested environments, RRM is a preferential approach for analyzing MET in maize breeding.

Impact of the Order of Legendre Polynomials in Random Regression Model on Genetic Evaluation for Milk Yield in Dairy Cattle Population.

The random regression test-day model has become the most commonly adopted model for routine genetic evaluations in dairy populations, which allows accurately accounting for genetic and environmental effects over lactation. The objective of this study was to explore appropriate random regression test-day models for genetic evaluation of milk yield in a Holstein population with a relatively small size, which is the common situation in regional or independent breeding companies to preform genetic evaluation. Data included 419,567 test-day records from 54,417 cows from the first lactation. Variance components and breeding values were estimated using a random regression test-day model with different orders (from first to fifth) of Legendre polynomials (LP) and accounted for homogeneous or heterogeneous residual variance across the lactation. Models were compared based on Akaike information criterion (AIC), Bayesian information criterion (BIC), and predictive ability. In general, models with a higher order of LP showed better goodness of fit based on AIC and BIC values. However, models with third, fourth, and fifth order of LP led to similar estimates of genetic parameters and predictive ability. Models with assumption of heterogeneous residual variances achieved better goodness of fit but yielded similar predictive ability compared with those with assumption of homogeneous residual variances. Therefore, a random regression model with third order of LP is suggested to be an appropriate model for genetic evaluation of milk yield in local Chinese Holstein populations.

Comparison of daughter's performance of progeny tested sires with pedigree selected sires in Holstein Friesian crossbred cattle

The present study was conducted to compare the estimated breeding values of progeny tested sires and pedigree selected sires for test day milk yield of crossbred Holstein Friesian cattle. First lactation milk yield records (1,20,198) of 12,971 daughters sired by 267 sires were collected from INAPH database maintained by NDDB. Variance and covariance components for test-day milk yield (TDMY) were estimated by different random regression test day models (RRTDM), viz. Cubic B- Spline function, Quadratic B-Spline function, Legendre polynomial (LP) function and Wilmink function using Average Information Restricted Maximum Likelihood (AIREML). Considering various criteria for comparison of different orders of TDMs, LP of 6th order for TDMY was the best fitted model for further estimation of breeding values. The heritability estimates ranged from 0.15 to 0.39 for TDMY using Leg_2 model. The additive genetic correlations were higher than the phenotypic correlations among different TDs. The additive genetic correlations between test day yields varied from 0.73 to 0.99. The expected progeny difference (EPD) for TDMYcalculated based on dam’s yield for the top ten and bottom ten PS bulls was 2,662.5 kg; whereas, the actual progeny difference (APD) for these bulls was -28.47 kg. While, EPD for top 10 and bottom 10 PT bulls selected based on EBVs was 2,820.52 kg whereas the APD for PT bulls was 890.48 kg. The difference in expected and actual MY of progeny was higher in PS bulls as compared to PT bulls indicating that the pedigree information for EPDs had a poor association with APDs and sire superiority is not reflected in progeny’s actual performance. The rank correlation between ranks of all PS and PT bulls were very poor and non-significant. The bulls selected based on estimated breeding values will give faster genetic progress and decision to select replacement bulls based on breeding values instead of dam’s yield will have positive effect on genetic progress.

The role of the intermediate state in angle-resolved photoelectron studies using (2 + 1) resonance-enhanced multiphoton ionization of the chiral terpenes, α-pinene and 3-carene

Photoelectron angular distributions (PADs), ranging up to the maximum 6th order Legendre polynomial term set by the Yang theorem, have been recorded for the (2 + 1) resonance enhanced multiphoton ionization (REMPI) of two terpene isomers, 3-carene and α-pinene, employing femtosecond lasers and electron velocity map imaging detection. PAD measurements made with coincident photoion detection allow ion fragmentation effects to be assessed. Using circular polarization and enantiomerically pure samples the PAD measurements are extended to include chiral (odd) Legendre polynomial terms, and these are analysed and discussed as multiphoton photoelectron circular dichroism (MP-PECD). Comparisons are also made with single photon (synchrotron radiation) PECD of these compounds. Although for a given compound a common final cation state is reached, pronounced differences are observed between PECD and MP-PECD, and between the alternative identified REMPI intermediate states in the case of MP-PECD.

First Principles Derivation of Displacement and Stress Function for Three-Dimensional Elastostatic Problems, and Application to the Flexural Analysis of Thick Circular Plates

In this study, stress and displacement functions of the three-dimensional theory of elasticity for homogeneous isotropic bodies are derived from first principles from the differential equations of equilibrium, the generalized stress – strain laws and the geometric relations of strain and displacement. It is found that the stress and displacement functions must be biharmonic functions. The derived functions are used to solve the elasticity problem of finding stresses and displacement fields in a thick circular plate with clamped edges for the case of uniformly distributed transverse load over the plate domain. Superposition of second to sixth order Legendre polynomials which are biharmonic functions are used in the thick circular plate problem as the stress function with the unknown constants as the parameters to be determined. Use of the stresses and displacement fields derived in terms of the stress and displacement function yielded the stress fields and displacement fields in terms of the unknown constants of the biharmonic stress function. Enforcement of the boundary conditions yielded the unknown constants, leading to a complete determination of the stress and displacement function for the stress fields and the displacement fields. The solutions obtained are comparable to solutions in the technical literature.

Random regression models for the evaluation of the growth of goats of the Anglonubian breed

ABSTRACT A total of 6593 weight records collected from 796 male and female Anglo-Nubian goats aged up to 130 days, offspring from 29 sires and 225 dams, were used to compare models and estimate genetic parameters throughout the growth curve by applying random regression models. Direct and maternal additive genetic effects and direct and maternal permanent environmental effects were included as random in the models. The contemporary groups were included as fixed effects and goat age at kidding was included as a covariable (linear and quadratic). The choice of the best model was based on the AIC, BIC and AICc criteria. Variance estimates of the four random effects increased as the animals aged. Direct heritability (h2) rose from 0.13 to 0.40 with age, whereas maternal heritability showed a low value. Genetic correlations of weight between closer ages were high. The most suitable random regression model to compare the fitting of random effects was that which employed the Legendre polynomials of quadratic order with homogeneous variance (3333-1).

Discontinuous Galerkin Spectral Element Method for Solving Population Balance Differential Equations

In this paper, we study the discontinuous Galerkin spectral element method for solving the population balance differential equation. We use the Legendre polynomials of order k as test functions on each element. Calculate the matrices used by Gauss Quadrature integration and then compare the numerical results with the other methods.

Estimation of genetic parameters for dairy traits and somatic cell score in the first 3 parities using a random regression test-day model in French Alpine goats

Lactation curve shape can affect an animal's health, feed requirements, and milk production throughout the year. We implemented a random regression model for the genetic evaluation of lactation curve shapes of dairy traits in French Alpine goats for their first 3 parities. Milk, fat, and protein yields, fat and protein contents, somatic cell score, and fat/protein ratio were considered. The data consisted of test-day records from 49,849 first lactation Alpine goats during their first 3 lactations. The reference model used a Legendre polynomial of order 2 for each parity to describe the genetic and permanent environmental effects, and was compared with a model that combined the second and third parities. A rank reduction of the variance-covariance matrix was also performed using an eigenvalue decomposition for each parity from the 2 models. Genetic parameters were consistent between the models tested. With a reduction to rank 2 and combining the second and third parities, the first 2 principal components correctly summarized the genetic variability of milk yield level and persistency, with a near-nil correlation between the 2, and with a much shorter computation time than the reference model. A favorable correlation of +0.43 between milk yield persistency and fat/protein ratio persistency at the beginning of the lactation was found from buck estimated breeding values.

Genetic evaluation of sow survival in Thailand commercial farms using random regression models

The aims of this study were 1) estimate heritability and variance components for sow survival traits using random regression model and 2) to identify the best model when conducting a sow survival genetic evaluation for Thailand commercial farms by comparing RRM with different covariance functions (Legendre polynomial and linear spline). A total of 11,595 and 11,336 sows from Landrace and Large White sows, respectively, were used to compare random regression models. The model using a second to third order Legendre polynomial for additive genetic effects and second to fourth order Legendre polynomial for permanent environmental effects (LG22, LG23, LG33 and LG34) and linear splines 3 to 4 knots for additive genetic effects and permanent environmental effects (SPL33 and SPL44) were used for analyses of genetic parameters. Bayesian interference using Gibbs sampling was used to estimate all covariance components. The model that included Legendre polynomial functions LG22 provided the lowest the deviance information criterion (DIC), provide the best fit for both the Landrace and Large White datasets. The heritability estimates for sow survival obtained with LG22 (the best fit model) ranged from 0.12 to 0.15 and 0.14 to 0.18 for Landrace and Large White sows, respectively. The genetic correlation among sow survival obtained with LG22 (the best fit model) ranged from 0.27 to 0.99, 0.43 to 0.99 for Landrace and Large White sows, respectively. Results from this study indicate that RRM could be used for genetic evaluation of sow survival.

Regularization of the electrostatics problem for three spheres and an electrostatic charge

A numerical-analytical algorithm for investigation of the potential of a sphere with a circular hole, surrounded by external and internal closed ribbon spheres, is constructed. The number of ribbons on the spheres is arbitrary. The ribbons on the spheres are separated by non-conductive, infinitely thin partitions. The partitions are located in planes parallel to the shear plane of the sphere with a hole. Each ribbon has its own independent potential. An electrostatic charge is placed between the outer sphere and the sphere with a hole in the axis of the structure. The full potential must satisfy, in particular, Maxwell’s equations, taking into account the absence of magnetic fields, satisfy the boundary conditions, have the required singularity at the point where the charge is placed. To solve this problem, we first used the method of partial domains and the method of separating variables in a spherical coordinate system. In this case, for the Fourier series, we use power functions and Legendre polynomials of integer orders. From the boundary conditions, using an auxiliary system of 3 equations with 4 unknowns, a pairwise system of functional equations of the first kind with respect to the coefficients of the Fourier series is obtained. The system is not effective for solving by direct methods. The method of inversion of the Volterra integral operator and semi-inversion of the matrix operators of the Dirichlet problem for the Laplace equation are applied. The method is based on the ideas of the analytical method of the Riemann - Hilbert problem. In this case, integral representations for the Legendre polynomials are used. A system of linear algebraic equations of the second kind with a compact matrix operator in the Hilbert space l`2 is obtained. The system is effectively solvable numerically for arbitrary parameters of the problem and analytically for the limiting parameters of the problem. Particular variants of the problem are considered.

Comparison of Different Order and Heterogeneous Residual Variances Legendre Polynomials in Random Regression Models

Comparison of Different Order and Heterogeneous Residual Variances Legendre Polynomials in Random Regression Models

Estimation of genetic parameters for mid-infrared–predicted lactoferrin and milk fat globule size in Holstein cattle

Lactoferrin (LF) and milk fat globule (MFG) are 2 biologically active components of milk with great economical and nutritional value in the dairy industry. The objectives of this study were to estimate (1) the heritability of mid-infrared (MIR)-predicted LF and MFG size (MFGS) and (2) the genetic correlations between predicted LF and MFGS with milk, fat, and protein yields, fat and protein percentages, and somatic cell score in first-parity Canadian Holstein cattle. A total of 109,029 test-day records from 22,432 cows and 1,572 farms for MIR-predicted LF and 109,212 test-day records from 22,424 cows and 1,559 farms for MIR-predicted MFGS were used in the analyses. Four separate 5-trait random regression test-day models were used. The models included days in milk, herd test date, and a polynomial regression on DIM nested in age-season of calving classes as fixed effects, random polynomial regressions on DIM nested in herd-year of calving, animal additive genetic and permanent environment classes, and a residual effect. Regression curves were modeled using orthogonal Legendre polynomials of order 4 for the fixed age-season of calving effect and of order 5 for the random effects. Moderate overall heritability estimates of 0.34 and 0.46 were estimated for the MIR-predicted LF and MIR-predicted MFGS, respectively. These heritability estimates were similar to the ones estimated for the direct measure of MFGS in a previous study. The genetic correlations between predicted MFGS and fat percentage (0.53) and between predicted LF and protein percentage (0.41) were both moderate and positive. Predicted LF and somatic cell score showed a weaker correlation (0.06) compared with other studies. The moderate genetic correlation between MIR-predicted MFGS and fat percentage and between MIR-predicted LF and protein percentage suggests that MIR predictions of MFGS and LF are not simply a function of the amount of fat and protein percentage, respectively, in the milk (i.e., the prediction equations are not simply predicting fat or protein percentages). Thus, these MIR-predicted values may provide additional information for selecting for fine milk components in Holstein cattle.

Random regression models for genetic analysis of body weight in crossbred pigs

Body weight of an animal is represented by a continuous function of time (longitudinal trait) and can be characterized by a trajectory with number of measurements. The present study was carried out to determine heritability estimates of body weight in crossbred pigs (75% Landrace + 25% Bareilly local) using random regression model with Legendre polynomials of quadratic power. Data of 9044 records of 1,292 crossbred piglets, progeny of 86 boars and 98 sows; born in 184 parities between 5 years from 2013–17 was used for the study. Records on weight at birth and at 1 week interval up to 6 week were used. Model included sex, year of birth, season of birth and parity as fixed effect, age of dam at farrowing as co-variable and direct additive genetic effect and maternal permanent environmental effect as random regression. There was a steady increase in body weight over the age from birth (0.96 kg) to 6th week (9.0 kg). Direct additive genetic (0.006 to 7.37 kg2), maternal permanent environment (0.053 to 70.07 kg2) and total phenotypic (0.18 to 77.56 kg2) variance increased continuously from birth to 6 week of age. In general, heritability estimates of body weight at different ages of pre-weaning stage were low ranging from 0.031 to 0.12. The estimate increased up to 1st week (0.119±0.025) with very low value at birth (0.031±0.015) and decreased thereafter to 0.095±0.022 at 6 week. Relative importance of each order of Legendre polynomials showed that quadratic Legendre polynomials with three regression coefficients were enough to capture almost all variability in the model to explain all additive genetic and maternal permanent environment variability. Hence, use of random regression model with quadratic Legendre polynomials was suggested for genetic analysis of pig data for growth.

Estimates of Genetic Parameters for Body Weight in Turkish Holstein Bulls using Random Regression Model

The objective of this study was to estimate genetic parameters for the body weights of Turkish Holstein bulls using the random regression model. The data set consists of 1475 body weight records from 395 Holstein bulls raised in the same herd. Body weight records of bulls aged between 32 and 725 days old were collected at approximately two-month intervals from December 2013 to October 2014. In the study body weight measurements made on the same day were accepted as a group and the bulls were grouped into 10 different groups according to their age. The additive genetic and permanent environmental effects were estimated using DFREML algorithm by third order Legendre polynomials. The additive genetic variance estimates ranged from 10.73 to 4867.07, the phenotypic variance estimates ranged from 382.84 to 5514.86 and permanent environmental variance estimates ranged from 0.33 to 63.27. The heritability values were estimated between 0.03 to 0.90. The phenotypic and additive genetic correlations between body weights were positively estimated between 0.085 to 0.89 and 0.53 to 0.94, respectively. It was concluded that use of body weight at an earlier age will give advantage in breeding studies for body weight at slaughter.

Use of Repeated Group Measurements with Drop Out Animals for Variance Component Estimation and Genetic Evaluation: A Simulation Study.

The efficiency of feed utilization plays an important role in animal breeding. However, measuring feed intake (FI) is costly on an individual basis under practical conditions. Using group measurements to model FI could be practically feasible and cost-effective. The objectives of this study were to develop a random regression model based on repeated group measurements with consideration of missing phenotypes caused by drop out animals. Focus is on variance components (VC) estimation and genetic evaluation, and to investigate the effect of group composition on VC estimation and genetic evaluation using simulated datasets. Data were simulated based on individual FI in a pig population. Each individual had measurement on FI at 6 different time points, reflecting 6 different weeks during the test period. The simulated phenotypes consisted of additive genetic, permanent environment, and random residual effects. Additive genetic and permanent environmental effects were both simulated and modeled by first order Legendre polynomials. Three grouping scenarios based on genetic relationships among the group members were investigated: (1) medium within and across pen genetic relationship; (2) high within group relationship; (3) low within group relationship. To investigate the effect of the drop out animals during test period, a proportion (15%) of animals with individual phenotypes was set as the drop out animals, and two drop out scenarios within each grouping scenario were assessed: (1) animals were randomly dropped out; (2) animals with lower phenotypes were dropped out based on the ranking at each time point. The results show that using group measurements yielded similar VCs estimates but with larger SDs compared with the corresponding scenario of using individual measurements. Compared to scenarios without drop out, similar VC estimates were observed when animals were dropped out randomly, whereas reduced VC estimates were observed when animals were dropped out by the ranking of phenotypes. Different grouping scenarios produced similar VC estimates. Compared to scenarios without drop out, there were no loss of accuracies of genetic evaluation for drop out scenarios. However, dropping out animals by the ranking of phenotypes produced larger bias of estimated breeding values compared to the scenario without dropped out animals and scenario of dropping out animals by random. In conclusion, with an optimized group structure, the developed model can properly handle group measurements with drop out animals, and can achieve comparable accuracy of genetic evaluation for traits measured at the group level.

A modified method for solving non-linear time and space fractional partial differential equations

PurposeThe purpose of the paper is to extend the differential quadrature method (DQM) for solving time and space fractional non-linear partial differential equations on a semi-infinite domain.Design/methodology/approachThe proposed method is the combination of the Legendre polynomials and differential quadrature method. The authors derived and constructed the new operational matrices for the fractional derivatives, which are used for the solutions of non-linear time and space fractional partial differential equations.FindingsThe fractional derivative of Lagrange polynomial is a big hurdle in classical DQM. To overcome this problem, the authors represent the Lagrange polynomial in terms of shifted Legendre polynomial. They construct a transformation matrix which transforms the Lagrange polynomial into shifted Legendre polynomial of arbitrary order. Then, they obtain the new weighting coefficients matrices for space fractional derivatives by shifted Legendre polynomials and use these in conversion of a non-linear fractional partial differential equation into a system of fractional ordinary differential equations. Convergence analysis for the proposed method is also discussed.Originality/valueMany engineers can use the presented method for solving their time and space fractional non-linear partial differential equation models. To the best of the authors’ knowledge, the differential quadrature method has never been extended or implemented for non-linear time and space fractional partial differential equations.

Calculation and verification of neutron irradiation damage with differential cross sections

The Displacement per Atom (DPA) rate is conventionally computed with DPA cross sections in reactor applications. The method of direct calculation with energy-angular distributions given in the Center of Mass (CM) frame is proposed and recommended in the present work. The methods for refining and verifying the calculations of DPA cross sections are proposed: (i) Gauss-Legendre-Quadrature-based Piecewise Integration (GLQPI) for ensuring the numeric convergence of integral over emission angle due to the discontinuity of integrand; (ii) verification of the convergence for trapezoidal integration over the secondary energy; (iii) interpolation of double-differential cross sections. For 56Fe of JEFF-3.1.1, the current numeric integration over emission angle is shown not convergent, whereas the direct trapezoidal over the secondary energy and the direct interpolation of energy-angle-integrated damage are shown accurate. On the other hand, it is shown that the DPA cross sections are overestimated if isotropic angular distributions are assumed. However, the DPA cross section is not sensitive to the high-order Legendre polynomials because the former is an angle-integrated quantity. Numerical results of neutron elastic scattering show that 2 orders of Legendre polynomials can give the DPA rates of 56Fe within 0.5% overestimation for fission reactors, while 4 orders are required for fusion reactors. For neutron inelastic scatterings-induced DPA, the first order Legendre polynomial is sufficient for both fission and fusion reactors.

Contribution of milk mid-infrared spectrum to improve the accuracy of test-day body weight predicted from stage, lactation number, month of test and milk yield

A regular and repeated recording of body weight (BW) is useful information for herd management. BW can be predicted regularly from animal characteristics such as age, lactation number, or lactation stage. Those traits are unfortunately animal unspecific. Adding animal specific information, which can be easily obtained on a large scale, to the BW prediction would be of utmost importance. There are good scientific reasons to suspect links between BW and animal specific characteristics, available in a repeated fashion, such as milk yield and milk composition. This study aimed to demonstrate the feasibility of predicting test-day BW from stage, lactation number, month of test, milk yield and mid-infrared spectra, representing milk composition. Five models were tested initially from 721 BW records collected in 6 herds: day in milk + number of lactation (equation 1a); equation1a + milk yield (equation 1b); only spectral data (equation 1c); equation 1c + equation 1a (equation 2); equation 2 + milk yield (equation 3). Then 3 other equations included the same explicative variables, except that the spectral data were regressed using second order Legendre Polynomials (PL) to take into account changes of spectral data within lactation. Equation 1a and 1b were built using linear regressions and equation 1c until 3 were built using partial least square regressions. These 3 last equations had a higher number of factors. Adding of MIR data in the equation increased of 7% the values of cross-validation R² (R²cv). Potential BW outliers were discarded using a residual analysis based on equation 3. From 662 records, the following statistical parameters were obtained: the calibration coefficient of determination (R²c) = 0.65, R²cv = 0.61, calibration root mean squared error of prediction (RMSEP)=38 kg, and RMSEPcv=40 kg. Low variation of R²c and RMSEPc values obtained from the herd validation confirmed the herd independence of predictions. However, large variability was observed for RMSEPv (37 to 64 kg) suggesting the need to increase the dataset in order to improve the robustness of the equation. By applying the equations on a large spectral database, it was confirmed that the addition of MIR data allows to better model the BW evolution within lactation. Based on these preliminary results, and if a larger validation confirms these findings, this approach could be used to develop equations that are better able to assess BW throughout lactation(s), BW being an important element for management and selection tools.