More animals than markers: a study into the application of the single step T-BLUP model in large-scale multi-trait Australian Angus beef cattle genetic evaluation

Multi-trait single step genetic evaluation is increasingly facing the situation of having more individuals with genotypes than markers within each genotype. This creates a situation where the genomic relationship matrix ($$\mathbf{G }$$ G ) is not of full rank and its inversion is algebraically impossible. Recently, the SS-T-BLUP method was proposed as a modified version of the single step equations, providing an elegant way to circumvent the inversion of the $$\mathbf{G }$$ G and therefore accommodate the situation described. SS-T-BLUP uses the Woodbury matrix identity, thus it requires an add-on matrix, which is usually the covariance matrix of the residual polygenic effet. In this paper, we examine the application of SS-T-BLUP to a large-scale multi-trait Australian Angus beef cattle dataset using the full BREEDPLAN single step genetic evaluation model and compare the results to the application of two different methods of using $$\mathbf{G }$$ G in a single step model. Results clearly show that SS-T-BLUP outperforms other single step formulations in terms of computational speed and avoids approximation of the inverse of $$\mathbf{G }$$ G .

Witnessing matrix identities and proof complexity

We use results from the theory of algebras with polynomial identities (PI-algebras) to study the witness complexity of matrix identities. A matrix identity of [Formula: see text] matrices over a field [Formula: see text]is a non-commutative polynomial (f(x1, …, xn)) over [Formula: see text], such that [Formula: see text] vanishes on every [Formula: see text] matrix assignment to its variables. For every field [Formula: see text]of characteristic 0, every [Formula: see text] and every finite basis of [Formula: see text] matrix identities over [Formula: see text], we show there exists a family of matrix identities [Formula: see text], such that each [Formula: see text] has [Formula: see text] variables and requires at least [Formula: see text] many generators to generate, where the generators are substitution instances of elements from the basis. The lower bound argument uses fundamental results from PI-algebras together with a generalization of the arguments in [P. Hrubeš, How much commutativity is needed to prove polynomial identities? Electronic colloquium on computational complexity, ECCC, Report No.: TR11-088, June 2011].We apply this result in algebraic proof complexity, focusing on proof systems for polynomial identities (PI proofs) which operate with algebraic circuits and whose axioms are the polynomial-ring axioms [P. Hrubeš and I. Tzameret, The proof complexity of polynomial identities, in Proc. 24th Annual IEEE Conf. Computational Complexity, CCC 2009, 15–18 July 2009, Paris, France (2009), pp. 41–51; Short proofs for the determinant identities, SIAM J. Comput. 44(2) (2015) 340–383], and their subsystems. We identify a decrease in strength hierarchy of subsystems of PI proofs, in which the [Formula: see text]th level is a sound and complete proof system for proving [Formula: see text] matrix identities (over a given field). For each level [Formula: see text] in the hierarchy, we establish an [Formula: see text] lower bound on the number of proof-steps needed to prove certain identities.Finally, we present several concrete open problems about non-commutative algebraic circuits and speed-ups in proof complexity, whose solution would establish stronger size lower bounds on PI proofs of matrix identities, and beyond.

Stable Subspaces of Positive Maps of Matrix Algebras

We study stable subspaces of positive extremal maps of finite dimensional matrix algebras that preserve trace and matrix identity (so-called bistochastic maps). We have established the existence of the isometric-sweeping decomposition for such maps. As the main result of the paper, we have shown that all extremal bistochastic maps acting on the algebra of matrices of size 3×3 fall into one of the three possible categories, depending on the form of the stable subspace of the isometric-sweeping decomposition. Our example of an extremal atomic positive map seems to be the first one that handles the case of that subspace being non-trivial. Lastly, we compute the entanglement witness associated with the extremal map and specify a large family of entangled states detected by it.

KETERKAITAN ANTAR SEKTOR DALAM PEREKONOMIAN DAERAH ISTIMEWA ACEH MELALUI ANALISIS TABEL I – O

Lingkage of economy between sectors in economic either economy of a country or a distric is very important to be analized. To know the economics interlated in Aceh special province, this research will observe about linkages between sectors through input – output table analizing. Data that used in this research was secondary data that taken from statistic center. The date were Table I-O 1988 and I-O 1997. Eight relative dominant sectors chosen, namely rice/padi, plantation, animal husbandry, food industry, drink, tobacco and oil and natural gas sectors. Model that used was input-output model. Variabel that used: matrix output domestic product (P), matrix identity (I), input matrix coeficient from domestic input (Ad) and final request matrix sector toward product (Fd). Based on the result explained that lingkages between sectors in special Aceh province economy having change for several years. Every sector had different lingkages stage with others sector that shown through the number of the input and output sector and the sum of input coeficient that required by that sector. Inplication of result mentioned that economic sectors will have changing and increasing if technology factor and coeficient stage of every economy sector predicted in every bussiness.Â

FAY'S MATRIX IDENTITY FOR VECTOR BUNDLES ON A CURVE

Among the less well-known identities established by Fay is a matrix identity which can be regarded as a vector bundle generalization of his better known trisecant identity. We show how our earlier proof of the latter can be adapted to prove his matrix identity.