scholarly journals Maximal Bootstrap Percolation Time on the Hypercube via Generalised Snake-in-the-Box

10.37236/7307 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Ivailo Hartarsky

In $r$-neighbour bootstrap percolation, vertices (sites) of a graph $G$ become "infected" in each round of the process if they have $r$ neighbours already infected. Once infected, they remain such. An initial set of infected sites is said to percolate if every site is eventually infected. We determine the maximal percolation time for $r$-neighbour bootstrap percolation on the hypercube for all $r \geq 3$ as the dimension $d$ goes to infinity up to a logarithmic factor. Surprisingly, it turns out to be $\frac{2^d}{d}$, which is in great contrast with the value for $r=2$, which is quadratic in $d$, as established by Przykucki (2012). Furthermore, we discover a link between this problem and a generalisation of the well-known Snake-in-the-Box problem.

1991 ◽  
Vol 1 (5) ◽  
pp. 685-692 ◽  
Author(s):  
Muhammad Sahimi ◽  
Tane S. Ray

ARCHALP ◽  
2018 ◽  
pp. 126-137
Author(s):  
Marina Hämmerle

We cannot understand the development of Vorarlberg’s architectural culture without its spatial, topographical, and socio-economic context. There is a great contrast between rural valleys and the busy, semi-urban Rhine Valley. With their exemplary buildings, states and municipalities model the production of excellent, contemporary architecture. Industrial and commercial architecture has achieved an impressive corporate identity as well. However, we rarely find the same quality in residential construction. Because of the high cost of real estate and construction apartment buildings have grown up like mushrooms, intruding upon areas formerly predominated by detached housing. Urban sprawl has eliminated the borders between the 29 municipalities of the Rhine Valley, resulting in a giant suburban landscape. To remedy this process, the players cooperate with the regional authorities as they carry out their vision of urban planning, including guidelines and ideas. Because planning and production have become so complex, urban and regional development has turned into an immense challenge. Provincial and municipal authorities value openness, participation, common good, ecology, and sustainability and involve citizens and adapt the process to their needs. Still, they must consider subsidy rules and regulations, which, until now, have privileged private property over common good and have prioritized ecological standards over architectural quality and the concerns of urban planning. Since 1997, the Vorarlberg Architecture Institute, has inspired, challenged, and spoken for the architectural-cultural scene. It continues to mediate and complement the discourse and activities of the Central Association of the Architects of Vorarlberg. In addition, the Chamber of Architects strives to improve competition procedures. The Energy Institute Vorarlberg supports ecology and promotes sustainability. The Quality Association “vorarlberger_holzbaukunst” has promoted the renaissance of timber construction. Carpenters and architects actively support the prefabrication and development of new technical solutions. Similarly, the members of the Werkraum Bregenzerwald, a craftsmen’s association, continue and transform the cultural heritage in sophisticated and resource-friendly ways, as evidenced by many buildings and the “Werkraumhaus” itself. Vorarlberg’s hospitality industry plays an important role in supporting and promoting the architectural culture. However, thoughtful and coordinated master planning is necessary to expand the quality of individual architectural projects to urban and regional planning and construction. This transition will be the most important challenge for the period of urban densification. Vorarlberg may be Alpine – even rural – but it is urban without doubt.[English translation by Ingeborg Fink].


2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Darina Dvinskikh ◽  
Alexander Gasnikov

Abstract We introduce primal and dual stochastic gradient oracle methods for decentralized convex optimization problems. Both for primal and dual oracles, the proposed methods are optimal in terms of the number of communication steps. However, for all classes of the objective, the optimality in terms of the number of oracle calls per node takes place only up to a logarithmic factor and the notion of smoothness. By using mini-batching technique, we show that the proposed methods with stochastic oracle can be additionally parallelized at each node. The considered algorithms can be applied to many data science problems and inverse problems.


1997 ◽  
Vol 34 (4) ◽  
pp. 898-907 ◽  
Author(s):  
Aihua Xia

This note gives the rate for a Wasserstein distance between the distribution of a Bernoulli process on discrete time and that of a Poisson process, using Stein's method and Palm theory. The result here highlights the possibility that the logarithmic factor involved in the upper bounds established by Barbour and Brown (1992) and Barbour et al. (1995) may be superfluous in the true Wasserstein distance between the distributions of a point process and a Poisson process.


1989 ◽  
Vol 22 (7) ◽  
pp. L297-L301 ◽  
Author(s):  
J Adler ◽  
D Stauffer ◽  
A Aharony

1983 ◽  
Vol 16 (32) ◽  
pp. 6263-6274 ◽  
Author(s):  
Y S Yang ◽  
Z Q Zhang

2015 ◽  
Vol 160 (5) ◽  
pp. 1249-1276 ◽  
Author(s):  
Tatyana S. Turova ◽  
Thomas Vallier

2003 ◽  
Vol 14 (04) ◽  
pp. 529-536 ◽  
Author(s):  
DIRK KURTSIEFER

The present article deals with the critical value pc of the three-dimensional bootstrap percolation. We will check the behavior of pc for different lengths of the lattice and additionally we will scale pc in the limit of an infinite lattice.


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