Critical cascades are found in many self-organizing systems. Here, we examine critical cascades as a design paradigm for logic and learning under the linear threshold model (LTM), and simple biologically inspired variants of it as sources of computational power, learning efficiency, and robustness. First, we show that the LTM can compute logic, and with a small modification, universal Boolean logic, examining its stability and cascade frequency. We then frame it formally as a binary classifier and remark on implications for accuracy. Second, we examine the LTM as a statistical learning model, studying benefits of spatial constraints and criticality to efficiency. We also discuss implications for robustness in information encoding. Our experiments show that spatial constraints can greatly increase efficiency. Theoretical investigation and initial experimental results also indicate that criticality can result in a sudden increase in accuracy.

Erdős asked if, for every pair of positive integers g and k, there exists a graph H having girth (H) = k and the property that every r-colouring of the edges of H yields a monochromatic cycle Ck. The existence of such graphs H was confirmed by the third author and Ruciński.

We consider the related numerical problem of estimating the order of the smallest graph H with this property for given integers r and k. We show that there exists a graph H on R10k2; k15k3 vertices (where R = R(Ck; r) is the r-colour Ramsey number for the cycle Ck) having girth (H) = k and the Ramsey property that every r-colouring of the edges of H yields a monochromatic Ck Two related numerical problems regarding arithmetic progressions in subsets of the integers and cliques in graphs are also considered.

The integration of Artificial Neural Networks (ANNs) and Feature Extraction (FE) in the context of the Sample- Partitioning Adaptive Reduced Chemistry approach was investigated in this work, to increase the on-the-fly classification accuracy for very large thermochemical states. The proposed methodology was firstly compared with an on-the-fly classifier based on the Principal Component Analysis reconstruction error, as well as with a standard ANN (s-ANN) classifier, operating on the full thermochemical space, for the adaptive simulation of a steady laminar flame fed with a nitrogen-diluted stream of n-heptane in air. The numerical simulations were carried out with a kinetic mechanism accounting for 172 species and 6,067 reactions, which includes the chemistry of Polycyclic Aromatic Hydrocarbons (PAHs) up to C$ {}_{20} $. Among all the aforementioned classifiers, the one exploiting the combination of an FE step with ANN proved to be more efficient for the classification of high-dimensional spaces, leading to a higher speed-up factor and a higher accuracy of the adaptive simulation in the description of the PAH and soot-precursor chemistry. Finally, the investigation of the classifier’s performances was also extended to flames with different boundary conditions with respect to the training one, obtained imposing a higher Reynolds number or time-dependent sinusoidal perturbations. Satisfying results were observed on all the test flames.

We present a polynomial-time Markov chain Monte Carlo algorithm for estimating the partition function of the antiferromagnetic Ising model on any line graph. The analysis of the algorithm exploits the ‘winding’ technology devised by McQuillan [CoRR abs/1301.2880 (2013)] and developed by Huang, Lu and Zhang [Proc. 27th Symp. on Disc. Algorithms (SODA16), 514–527]. We show that exact computation of the partition function is #P-hard, even for line graphs, indicating that an approximation algorithm is the best that can be expected. We also show that Glauber dynamics for the Ising model is rapidly mixing on line graphs, an example being the kagome lattice.

Facilitation style appears to be an important determinant of design team effectiveness. The neutrality of the group facilitator may be a key factor; however, the characteristics and impact of neutrality are relatively understudied. In a designed classroom setting, we examine the impact of two different approaches to group facilitation: (i) facilitator’s neutrality expressed as low equidistance and high impartiality and (ii) facilitator’s neutrality expressed as high equidistance and low impartiality, on team trust, trust to the facilitator and team potency. To do this, we conducted a repeated-measures experiment with a student sample. Our results indicate that facilitators expressing neutrality through low equidistance and high impartiality had a greater positive impact on team trust. The two approaches did not differ on team potency and facilitator trust. These results contribute to developing theories of design facilitation and team effectiveness by suggesting how facilitation may shape team trust and potency in group design. Based on our findings, we point to the need for future work to further examine the impact of facilitator’s process awareness and neutrality, and show how facilitation methods may benefit teams during creative design teamwork.

Prototyping constitutes a major theme of design education and an integral part of engineering design academic courses. Physical prototypes and the model building process, in particular, have been proved to boost students’ creativity and resourcefulness and assist in the better evaluation of concepts. However, students’ usage of prototypes has still not been explored in depth with the aim of being transformed into educational guidelines. This paper presents an investigation of students’ reasoning behind prototyping activities based on the concept of Purposeful Prototyping, developed in the authors’ previous work. This is performed by identifying instances of prototype use in students’ design projects and by discovering which types of prototyping purposes they apply and to what extent, as well as by studying the relationships between purposes, early design stages, academic performance and project planning. The analysis of the results shows that prototyping can support students’ learning objectives by acting as a project scheduling tool and highlights the contribution of early-stage prototyping in academic performance. It is also confirmed that students’ limited prototyping scope prevents them from gaining prototyping’s maximum benefits and that they require strategic guidelines tailored to their needs. A new, improved list of prototyping purposes is proposed based on the study’s results.

We prove that if $A \subseteq [X,\,2X]$ and $B \subseteq [Y,\,2Y]$ are sets of integers such that gcd (a, b) ⩾ D for at least δ|A||B| pairs (a, b) ε A × B then $|A||B|{ \ll _{\rm{\varepsilon }}}{\delta ^{ - 2 - \varepsilon }}XY/{D^2}$. This is a new result even when δ = 1. The proof uses ideas of Koukoulopoulos and Maynard and some additional combinatorial arguments.