Extremal dynamics represents a path to self-organized criticality in which the order parameter is tuned to a value of zero. The order parameter is associated with a phase transition to an absorbing state. Given a process that exhibits a phase transition to an absorbing state, we define an "extremal absorbing" process, providing the link to the associated extremal (nonabsorbing) process. Station...

Consider a polygon P and all neighboring circles (circles going through three consecutive vertices of P ). We say that a neighboring circle is extremal if it is empty (no vertices of P inside) or full (no vertices of P outside). It is well known that for any convex polygon there exist at least two empty and at least two full circles, i.e. at least four extremal circles. In 1990 Schatteman consi...

The blow-up of a graph H is the graph obtained from replacing each edge in H by a clique of the same size where the new vertices of the cliques are all different. Erdős et al. and Chen et al. determined the extremal number of blow-ups of stars. Glebov determined the extremal number and found all extremal graphs for blowups of paths. We determine the extremal number and find the extremal graphs ...

We study the Sasaki cone of a CR structure of Sasaki type on a given closed manifold. We introduce an energy functional over the cone, and use its critical points to single out the strongly extremal Reeb vectors fields. Should one such vector field be a member of the extremal set, the scalar curvature of a Sasaki extremal metric representing it would have the smallest L2-norm among all Sasakian...

The joint spectral radius of a bounded set of d × d real matrices is defined to be the maximum possible exponential growth rate of products of matrices drawn from that set. For a fixed set of matrices, a sequence of matrices drawn from that set is called extremal if the associated sequence of partial products achieves this maximal rate of growth. An influential conjecture of J. Lagarias and Y. ...

This is a set of notes that is basically and expanded version of the paper Extremal Approximately Convex Functions and Estimating the Size of Convex Hulls. The differences are a few extra pictures, Section 2.7 which is an exposition of results of Ng and Nikodem [5] about measurable approximately convex functions, and an alternate proof of Theorem 2.27 is included. Contents 1. Introduction 2 2. ...

We investigate extremal statistical properties such as the maximal and the minimal heights of randomly generated binary trees. By analyzing the master evolution equations we show that the cumulative distribution of extremal heights approaches a traveling wave form. The wave front in the minimal case is governed by the small-extremal-height tail of the distribution, and conversely, the front in ...

A code C is called formally self-dual if C and C⊥ have the same weight enumerators. There are four types of nontrivial divisible formally self-dual codes over F2, F3, and F4. These codes are called extremal if their minimum distances achieve the MallowsSloane bound. S. Zhang gave possible lengths for which extremal self-dual codes do not exist. In this paper, we define near-extremal formally se...