Nature versus built

Nature is made up of various living and non-living things which is related to plants, animals and other features such as mountains, deserts and seas. They are connected and some of the relationships between members are direct and obvious and ecosystem occurred to balance the amount of living things. Built environment is referring to aspects of creature human-made surroundings. Human activities caused some environmental issues and disturbance the nature. However, the technologies introduce sustainability building and creature as conservation our nature. There also some ways can be practice at house to save the environment

Symmetry of tilings of the plane

We discuss two new results on tilings of the plane. In the first, we give sufficient conditions for the tilings associated with an inflation rule to be uniquely ergodic under translations, the conditions holding for the pinwheel inflation rule. In the second result we prove there are matching rules for the pinwheel inflation rule, making the system the first known to have complete rotational symmetry.Comment: 5 page

Optimally dense packings of hyperbolic space

In previous work a probabilistic approach to controlling difficulties of density in hyperbolic space led to a workable notion of optimal density for packings of bodies. In this paper we extend an ergodic theorem of Nevo to provide an appropriate definition of optimal dense packings. Examples are given to illustrate various aspects of the density problem, in particular the shift in emphasis from the analysis of individual packings to spaces of packings.Comment: 27 pages, 11 figure

Phase transitions in exponential random graphs

We derive the full phase diagram for a large family of two-parameter exponential random graph models, each containing a first order transition curve ending in a critical point.Comment: Published in at http://dx.doi.org/10.1214/12-AAP907 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org