Gaussian graphical models are useful tools for conditional independence structure inference of multivariate random variables. Unfortunately, Bayesian inference of latent graph structures is challenging due to exponential growth of $\mathcal{G}_n$, the set of all graphs in n vertices. One approach that has been proposed to tackle this problem is to limit search to subsets of $\mathcal{G}_n$. In this paper we study subsets that are vector subspaces with the cycle space $\mathcal{C}_n$ as the main example. We propose a novel prior on $\mathcal{C}_n$ based on linear combinations of cycle basis elements and present its theoretical properties. Using this prior, we implement a Markov chain Monte Carlo algorithm, and show that (i) posterior edge inclusion estimates computed with our technique are comparable to estimates from the standard technique despite searching a smaller graph space, and (ii) the vector space perspective enables straightforward implementation of MCMC algorithms.

In set theory without the Axiom of Choice ( $\mathsf {AC}$ ), we investigate the open problem of the deductive strength of statements which concern the existence of almost disjoint and maximal almost disjoint (MAD) families of infinite-dimensional subspaces of a given infinite-dimensional vector space, as well as the extension of almost disjoint families in infinite-dimensional vector spaces to MAD families.

In this paper, we study the relation of the size of the class two quotients of a linear group and the size of the vector space. We answer a question raised in Keller and Yang [Class 2 quotients of solvable linear groups, J. Algebra 509 (2018), 386-396].

This paper serves as a short overview of the JNLE special issue on representation of the meaning of the sentence, bringing together traditional symbolic and modern continuous approaches. We indicate notable aspects of sentence meaning and their compatibility with the two streams of research and then summarize the papers selected for this special issue.

We propose a fundamentally new approach to Datalog evaluation. Given a linear Datalog program DB written using N constants and binary predicates, we first translate if-and-only-if completions of clauses in DB into a set E q (DB) of matrix equations with a non-linear operation, where relations in M DB, the least Herbrand model of DB, are encoded as adjacency matrices. We then translate E q (DB) into another, but purely linear matrix equations Ẽ q (DB). It is proved that the least solution of Ẽ q (DB) in the sense of matrix ordering is converted to the least solution of E q (DB) and the latter gives M DB as a set of adjacency matrices. Hence, computing the least solution of Ẽ q (DB) is equivalent to computing M DB specified by DB. For a class of tail recursive programs and for some other types of programs, our approach achieves O(N 3) time complexity irrespective of the number of variables in a clause since only matrix operations costing O(N 3) or less are used. We conducted two experiments that compute the least Herbrand models of linear Datalog programs. The first experiment computes transitive closure of artificial data and real network data taken from the Koblenz Network Collection. The second one compared the proposed approach with the state-of-the-art symbolic systems including two Prolog systems and two ASP systems, in terms of computation time for a transitive closure program and the same generation program. In the experiment, it is observed that our linear algebraic approach runs 101 ~ 104 times faster than the symbolic systems when data is not sparse. Our approach is inspired by the emergence of big knowledge graphs and expected to contribute to the realization of rich and scalable logical inference for knowledge graphs.

We study domination in zero-divisor graphs of matrix rings over a commutative ring with 1.