Study of the interactions in a three-way crossed classification model

Crossed classification models are applied in many investigations taking in consideration the existence of interaction between all factors or, in alternative, excluding all interactions, and in this case only the effects and the error term are considered. In this work we use commutative Jordan algebras in the study of the algebraic structure of these designs and we use them to obtain similar designs where only some of the interactions are considered. We finish presenting the expressions of the variance componentes estimators.

Algebraic structure for interaction on mixed models

Binary operations on commutative Jordan algebras, CJA, can be used to study interactions between sets of factors belonging to a pair of models in which one nests the other. It should be noted that from two CJA we can, through these binary operations, build CJA. So when we nest the treatments from one model in each treatment of another model, we can study the interactions between sets of factors of the first and the second models.

Orthogonal models, structure crossing, nesting and inference

We intend to study the algebraic structure of the simple orthogonal models to use them, through binary operations as building blocks in the construction of more complex orthogonal models. We start by presenting some matrix results considering Commutative Jordan Algebras of symmetric matrices, CJAs. Next, we use these results to study the algebraic structure of orthogonal models, obtained by crossing and nesting simpler ones. Then, we study the normal models with OBS, which can also be orthogonal models. We intend to study normal models with OBS (Orthogonal Block Structure), NOBS (Normal Orthogonal Block Structure), obtaining condition for having complete and suffcient statistics, having UMVUE, is unbiased estimators with minimal covariance matrices whatever the variance components. Lastly, see ([Pereira et al. (2014)]), we study the algebraic structure of orthogonal models, mixed models whose variance covariance matrices are all positive semi definite, linear combinations of known orthogonal pairwise orthogonal projection matrices, OPOPM, and whose least square estimators, LSE, of estimable vectors are best linear unbiased estimator, BLUE, whatever the variance components, so they are uniformly BLUE, UBLUE. From the results of the algebraic structure we will get explicit expressions for the LSE of these models.

Strongly non-degenerate Lie algebras

Let A be a semiprime 2 and 3-torsion free non-commutative associative algebra. We show that the Lie algebra Der(A) of(associative) derivations of A is strongly non-degenerate, which is a strong form of semiprimeness for Lie algebras, under some additional restrictions on the center of A. This result follows from a description of the quadratic annihilator of a general Lie algebra inside appropriate Lie overalgebras. Similar results are obtained for an associative algebra A with involution and the Lie algebra SDer(A) of involution preserving derivations of A

Orthogonal systems in finite grahps

To a finite graph there corresponds a free partially commutative group: with the given graph as commutation graph. In this paper we construct an orthogonality theory for graphs and their corresponding free partially commutative groups. The theory developed here provides tools for the study of the structure of partially commutative groups, their universal theory and automorphism groups. In particular the theory is applied in this paper to the centraliser lattice of such groups.

Localization of algebras over coloured operads

We give sufficient conditions for homotopical localization functors to preserve algebras over coloured operads in monoidal model categories. Our approach encompasses a number of previous results about preservation of structures under localizations, such as loop spaces or infinite loop spaces, and provides new results of the same kind. For instance, under suitable assumptions, homotopical localizations preserve ring spectra (in the strict sense, not only up to homotopy), modules over ring spectra, and algebras over commutative ring spectra, as well as ring maps, module maps, and algebra maps. It is principally the treatment of module spectra and their maps that led us to the use of coloured operads (also called enriched multicategories) in this context.

Higher arithmetic Chow groups

We give a new construction of higher arithmetic Chow groups for quasi-projective arithmetic varieties over a field. Our definition agrees with the higher arithmetic Chow groups defined by Goncharov for projective arithmetic varieties over a field. These groups are the analogue, in the Arakelov context, of the higher algebraic Chow groups defined by Bloch. The degree zero group agrees with the arithmetic Chow groups of Burgos. Our new construction is shown to be a contravariant functor and is endowed with a product structure, which is commutative and associative.

Dealing with clients' diversity in test situations: Client categorisations in psychologists' accounts of their practices

Reducing a test administration to standardised procedures reflects the test designers' standpoint. However, from the practitioners' standpoint, each client is unique. How do psychologists deal with both standardised test administration and clients' diversity? To answer this question, we interviewed 17 psychologists working in three public services for children and adolescents about their assessment practices. We analysed the numerous "client categorisations" they produced in their accounts. We found that they had shared perceptions about their clients' diversity, and reported various non-standard practices that complemented standardised test administration, but also differed from them or were even forbidden. They seem to experience a dilemma between: (a) prescribed and situated practices; (b) scientific and situated reliability; (c) commutative and distributive justice. For practitioners, dealing with clients' diversity this is a practical problem, halfway between a problem-solving task and a moral dilemma.

Macaulay style formulas for sparse resultants

We present formulas for computing the resultant of sparse polyno- mials as a quotient of two determinants, the denominator being a minor of the numerator. These formulas extend the original formulation given by Macaulay for homogeneous polynomials.

On the depth of blowup algebras of ideals with analytic deviation one

Let I be an ideal in a local Cohen-Macaulay ring (A, m). Assume I to be generically a complete intersection of positive height. We compute the depth of the Rees algebra and the form ring of I when the analytic deviation of I equals one and its reduction number is also at most one. The formu- las we obtain coincide with the already known formulas for almost complete intersection ideals.

Bayes linear spaces

Linear spaces consisting of σ-finite probability measures and infinite measures (improper priors and likelihood functions) are defined. The commutative group operation, called perturbation, is the updating given by Bayes theorem; the inverse operation is the Radon-Nikodym derivative. Bayes spaces of measures are sets of classes of proportional measures. In this framework, basic notions of mathematical statistics get a simple algebraic interpretation. For example, exponential families appear as affine subspaces with their sufficient statistics as a basis. Bayesian statistics, in particular some well-known properties of conjugated priors and likelihood functions, are revisited and slightly extended