Topological finiteness properties of monoids. Part 1: Foundations

We initiate the study of higher dimensional topological finiteness properties of monoids. This is done by developing the theory of monoids acting on CW complexes. For this we establish the foundations of $M$-equivariant homotopy theory where $M$ is a discrete monoid. For projective $M$-CW complexes we prove several fundamental results such as the homotopy extension and lifting property, which we use to prove the $M$-equivariant Whitehead theorems. We define a left equivariant classifying space as a contractible projective $M$-CW complex. We prove that such a space is unique up to $M$-homotopy equivalence and give a canonical model for such a space via the nerve of the right Cayley graph category of the monoid. The topological finiteness conditions left-$\mathrm{F}_n$ and left geometric dimension are then defined for monoids in terms of existence of a left equivariant classifying space satisfying appropriate finiteness properties. We also introduce the bilateral notion of $M$-equivariant classifying space, proving uniqueness and giving a canonical model via the nerve of the two-sided Cayley graph category, and we define the associated finiteness properties bi-$\mathrm{F}_n$ and geometric dimension. We explore the connections between all of the these topological finiteness properties and several well-studied homological finiteness properties of monoids which are important in the theory of string rewriting systems, including $\mathrm{FP}_n$, cohomological dimension, and Hochschild cohomological dimension. We also develop the corresponding theory of $M$-equivariant collapsing schemes (that is, $M$-equivariant discrete Morse theory), and among other things apply it to give topological proofs of results of Anick, Squier and Kobayashi that monoids which admit presentations by complete rewriting systems are left-, right- and bi-$\mathrm{FP}_\infty$.Comment: 59 pages, 1 figur

Speeding up neighborhood search in local Gaussian process prediction

Recent implementations of local approximate Gaussian process models have pushed computational boundaries for non-linear, non-parametric prediction problems, particularly when deployed as emulators for computer experiments. Their flavor of spatially independent computation accommodates massive parallelization, meaning that they can handle designs two or more orders of magnitude larger than previously. However, accomplishing that feat can still require massive supercomputing resources. Here we aim to ease that burden. We study how predictive variance is reduced as local designs are built up for prediction. We then observe how the exhaustive and discrete nature of an important search subroutine involved in building such local designs may be overly conservative. Rather, we suggest that searching the space radially, i.e., continuously along rays emanating from the predictive location of interest, is a far thriftier alternative. Our empirical work demonstrates that ray-based search yields predictors with accuracy comparable to exhaustive search, but in a fraction of the time - bringing a supercomputer implementation back onto the desktop.Comment: 24 pages, 5 figures, 4 table

The Laws of Unintended Consequence: The Effect of Labour Legislation on Wages and Strikes

When politicians consider intervening in labour disputes, they should also consider the long-term, potentially unintended results of such action. In this study, the authors investigate the lessons from previous government legislative interventions, whether through compulsory arbitration, “back-to-work” legislation or bans on replacement workers during strikes, and find these actions have unintended results that give reason for sober second thought.Economic Growth and Innovation, Canadian federal and provincial governments, labour relations, compulsory arbitration, back-to-work legislation

Self tolerance in a minimal model of the idiotypic network

We consider the problem of self tolerance in the frame of a minimalistic model of the idiotypic network. A node of this network represents a population of B lymphocytes of the same idiotype which is encoded by a bit string. The links of the network connect nodes with (nearly) complementary strings. The population of a node survives if the number of occupied neighbours is not too small and not too large. There is an influx of lymphocytes with random idiotype from the bone marrow. Previous investigations have shown that this system evolves toward highly organized architectures, where the nodes can be classified into groups according to their statistical properties. The building principles of these architectures can be analytically described and the statistical results of simulations agree very well with results of a modular mean field theory. In this paper we present simulation results for the case that one or several nodes, playing the role of self, are permanently occupied. We observe that the group structure of the architecture is very similar to the case without self antigen, but organized such that the neighbours of the self are only weakly occupied, thus providing self tolerance. We also treat this situation in mean field theory which give results in good agreement with data from simulation.Comment: 7 pages, 6 figures, 1 tabl

Ruthenium Olefin Metathesis Catalysts Bearing Carbohydrate-Based N-Heterocyclic Carbenes

Ru-based olefin metathesis catalysts containing carbohydrate-derived NHCs from glucose and galactose were synthesized and characterized by NMR spectroscopy. 2D-NMR spectroscopy revealed the presence of Ru−C (benzylidene) rotamers at room temperature, and the rate of rotation was measured using magnetization transfer and VT-NMR spectroscopy. The catalysts were found to be effective at ring-opening metathesis polymerization (ROMP), ring-closing metathesis (RCM), cross-metathesis (CM), and asymmetric ring-opening cross-metathesis (AROCM) and showed surprising selectivity in both CM and AROCM