When Are Torsionless Modules Projective?

In this paper, we study the problem when a finitely generated torsionless module is projective. Let $\Lambda$ be an Artinian local algebra with radical square zero. Then a finitely generated torsionless $\Lambda$-module $M$ is projective if ${\rm Ext^1_\Lambda}(M,M)=0$. For a commutative Artinian ring $\Lambda$, a finitely generated torsionless $\Lambda$-module $M$ is projective if the following conditions are satisfied: (1) ${\rm Ext}^i_{\Lambda}(M,\Lambda)=0$ for $i=1,2,3$; and (2) ${\rm Ext}^i_{\Lambda}(M,M)=0$ for $i=1,2$. As a consequence of this result, we have that for a commutative Artinian ring $\Lambda$, a finitely generated Gorenstein projective $\Lambda$-module is projective if and only if it is selforthogonal.Comment: 10 page

A Bayesian Approach toward Active Learning for Collaborative Filtering

Collaborative filtering is a useful technique for exploiting the preference patterns of a group of users to predict the utility of items for the active user. In general, the performance of collaborative filtering depends on the number of rated examples given by the active user. The more the number of rated examples given by the active user, the more accurate the predicted ratings will be. Active learning provides an effective way to acquire the most informative rated examples from active users. Previous work on active learning for collaborative filtering only considers the expected loss function based on the estimated model, which can be misleading when the estimated model is inaccurate. This paper takes one step further by taking into account of the posterior distribution of the estimated model, which results in more robust active learning algorithm. Empirical studies with datasets of movie ratings show that when the number of ratings from the active user is restricted to be small, active learning methods only based on the estimated model don't perform well while the active learning method using the model distribution achieves substantially better performance.Comment: Appears in Proceedings of the Twentieth Conference on Uncertainty in Artificial Intelligence (UAI2004

The Wild Bootstrap for the Variance Radio Test

Rapport de rechercheNuméro de référence interne originel : a1.1 g 105

An extremal problem on group connectivity of graphs

Let A be an Abelian group, n \u3e 3 be an integer, and ex(n, A) be the maximum integer such that every n-vertex simple graph with at most ex(n, A) edges is not A-connected. In this paper, we study ex(n, A) for IAI \u3e 3 and present lower and upper bounds for 3 \u3c IAI 5. 0 2012 Elsevier Ltd. All rights reserved

Edge coloring of simple graphs and edge -face coloring of simple plane graphs

We prove that chie( G) = Delta if Delta ≥ 5 and g ≥ 4, or Delta ≥ 4 and g ≥ 5, or Delta ≥ 3 and g ≥ 9. In addition, if chi(Sigma) \u3e 0, then chie( G) = Delta if Delta ≥ 3 and g ≥ 8 where Delta, g is the maximum degree, the girth of the graph G, respectively.;It is proved that G is not critical if d¯ ≤ 6 and Delta ≥ 8, or d¯ ≤ 203 and Delta ≥ 9. This result generalizes earlier results.;Given a simple plane graph G, an edge-face k-coloring of G is a function &phis; : E(G) ∪ F(G) {lcub}1, ···, k{rcub} such that, for any two adjacent elements a, b ∈ E(G) ∪ F(G), &phis;( a) ≠ &phis;(b). Denote chie( G), chief(G), Delta( G) the edge chromatic number, the edge-face chromatic number and the maximum degree of G, respectively. We prove that chi ef(G) = chie( G) = Delta(G) for any 2-connected simple plane graph G with Delta(G) ≥ 24

Estimation for Zero-Inflated Beta-Binomial Regression Model with Missing Response and Covariate Measurement Error

Discrete, binary data with over-dispersion and zero-inflation can arise in toxicology and other similar fields. In studies where the litter is an experimental unit, there is a ``litter effect which means that the litter mates respond more alike than animals from other litters. In experimental data, foetuses in the same litter have similar responses to the treatment. The probability of ``success may not remain constant throughout the litters. In regression analysis of such data another problem that may arise in practice is that some responses may be missing or/and some covariates may have measurement error. In this dissertation we develop an estimation procedure for the parameters of a zero-inflated over-dispersed binomial model in the presence of missing responses without/with considering covariate measurement errors. A weighted expectation maximization algorithm is used for the maximum likelihood (ML) estimation of the parameters involved. Extensive simulations are conducted to study the properties of the estimates in terms of average estimates (AE), relative bias (RB), variance (VAR), mean squared error (MSE) and coverage probability (CP) of estimates. Simulations show much superior properties of the estimates obtained using the weighted expectation maximization algorithm. Some illustrative examples and a discussion are given