Generalized loopy 2U: A new algorithm for approximate inference in credal networks

Credal networks generalize Bayesian networks by relaxing the requirement of precision of probabilities. Credal networks are considerably more expressive than Bayesian networks, but this makes belief updating NP-hard even on polytrees. We develop a new efficient algorithm for approximate belief updating in credal networks. The algorithm is based on an important representation result we prove for general credal networks: that any credal network can be equivalently reformulated as a credal network with binary variables; moreover, the transformation, which is considerably more complex than in the Bayesian case, can be implemented in polynomial time. The equivalent binary credal network is then updated by L2U, a loopy approximate algorithm for binary credal networks. Overall, we generalize L2U to non-binary credal networks, obtaining a scalable algorithm for the general case, which is approximate only because of its loopy nature. The accuracy of the inferences with respect to other state-of-the-art algorithms is evaluated by extensive numerical tests.     

Nilpotency of group ring units symmetric with respect to an involution

Let F be an infinite field of characteristic different from 2. Let G be a torsion group having an involution ∗, and consider the units of the group ring FG that are symmetric with respect to the induced involution. We classify the groups G such that these symmetric units satisfy a nilpotency identity (x₁,…,x_n)=1.     

Strong commutativity preserving maps in prime rings with involution

Let A be a prime ring of characteristic not 2, with center Z(A) and with involution *. Let S be the set of symmetric elements of A. Suppose that f:S→A is an additive map such that [f(x),f(y)]=[x,y] for all x,y∈S. Then unless A is an order in a 4-dimensional central simple algebra, there exists an additive map μ:S→Z(A) such that f(x)=x+μ(x) for all x∈S or f(x)=-x+μ(x) for all x∈S.     

Which finite simple groups are unit groups?

We prove that if G$G$ is a finite simple group which is the unit group of a ring, then G$G$ is isomorphic to: (a) a cyclic group of order 2; or (b) a cyclic group of prime order 2^k−1$2^k-1$ for some k$k$; or (c) a projective special linear group _PSLn(_F2)${PSL}_n\left({\mathbb{F}}_2\right)$ for some n≥3$n\ge 3$. Moreover, these groups do all occur as unit groups. We deduce this classification from a more general result, which holds for groups G$G$ with no non-trivial normal 2-subgroup.     

Penultimate limiting forms in extreme value theory

Summary Let {X _n}_n≧1 be a sequence of independent, identically distributed random variables. If the distribution function (d.f.) ofM _n=max (X ₁,…,X _n), suitably normalized with attraction coefficients {α_n}_n≧1(α_n>0) and {b _n}_n≧1, converges to a non-degenerate d.f.G(x), asn→∞, it is of interest to study the rate of convergence to that limit law and if the convergence is slow, to find other d.f.'s which better approximate the d.f. of(M _n−b_n)/a_n thanG(x), for moderaten. We thus consider differences of the formF ⁿ(a_nx+b_n)−G(x), whereG(x) is a type I d.f. of largest values, i.e.,G(x)≡Λ(x)=exp (-exp(−x)), and show that for a broad class of d.f.'sF in the domain of attraction of Λ, there is a penultimate form of approximation which is a type II [Ф _α(x)=exp (−x^−α), x>0] or a type III [Ψ _α(x)= exp (−(−x)^α), x<0] d.f. of largest values, much closer toF ⁿ(a_nx+b_n) than the ultimate itself.     

A New Look at Mourre’s Commutator Theory

Mourre’s commutator theory is a powerful tool to study the continuous spectrum of self-adjoint operators and to develop scattering theory. We propose a new approach of its main result, namely the derivation of the limiting absorption principle (LAP) from a so called Mourre estimate. We provide a new interpretation of this result. Received: September 20, 2006. Revised: February 1, 2007.     

Metric spaces with discrete topological fundamental group

With a certain natural topology, the fundamental group of a locally path connected metric space X becomes discrete if and only if X is semilocally simply connected. We include some related material and offer a counterexample to a similar result.     

A completion theorem for pro-G-spectra

We give a very general completion theorem for pro-spectra. We show that, if G is a compact Lie group, M[∗] is a pro-G-spectrum, and F is a family of (closed) subgroups of G, then the mapping pro-spectrum F(EF₊,M[∗]) is the F-adic completion of M[∗], in the sense that the map M[∗]→F(EF₊,M[∗]) is the universal map into an algebraically F-adically complete pro-spectrum. Here, F(EF₊,M[∗]) denotes the pro-G-spectrum , where runs over the finite subcomplexes of EF₊.     

Commutativity conditions for rings: 1950–2005

During the last 55 years there have been many results concerning conditions that force a ring to be commutative. These results were stimulated by Jacobson's famous result and were extensively developed by Herstein. This paper will survey the area by organizing the results according to whether they come from variations on Herstein's conditions, depend on general polynomial conditions, depend on the presence of a derivation, or whether a ring has special properties that make commutativity more easily accessible. Finally, the most recent conditions concern product sets and lead to results in a new area of inquiry.