We show that the longest k-alternating substring of a random permutation has length asymptotic to 2(n− k)/3.

This paper studies the fundamental limits of the minimum average length of variable-length compression when a nonzero error probability ǫ is tolerated. We give non-asymptotic bounds on the minimum average length in terms of Erokhin’s rate-distortion function and we use those bounds to obtain a Gaussian approximation on the speed of approach to the limit which is quite accurate for all but small...

Exploiting a relationship between closed geodesics on a generic closed hyperbolic surface S and a certain unipotent flow on the product space T 1 (S) × T 1 (S), we obtain a local asymptotic equidistribution result for long closed geodesics on S. Applications include asymptotic estimates for the number of pants immersions into S satisfying various geometric constraints. Also we show that two clo...

Exploiting a relationship between closed geodesics on a generic closed hyperbolic surface S and a certain unipotent flow on the product space T 1 (S) × T 1 (S), we obtain a local asymptotic equidistribution result for long closed geodesics on S. Applications include asymptotic estimates for the number of pants immersions into S satisfying various geometric constraints. Also we show that two clo...

Weyl's expansion for the asymptotic mode density of billiards consists of the area, length, curvature, and corner terms. The area term has been associated with the so-called zero-length orbits. Here, closed nonperiodic paths corresponding to the length and corner terms are constructed.

In this paper, we consider the on-line scheduling of two parallel identical machines sharing a single server with the objective of minimizing the latest completion time of all jobs. Each job has to be setup by the server before being processed on one of the machines. Three special cases: equal length jobs, equal processing times and regular equal setup times are considered and the asymptotic co...

We analyse the average number of buckets in a Linear Bucket tree created by n points uniformly dispatched on an interval of length y. A new bucket is created when a point does not fall in an existing bucket. The bucket is the interval of length 2 centered on the point. We illustrate this concept by an interesting tale of how the moon’s surface took on its present form. Thanks to an explicit Lap...

Analytic solution for the average path length in a large class of uncorrelated random networks with hidden variables is found. We apply the approach to classical random graphs of Erdös and Rényi (ER), evolving networks introduced by Barabási and Albert as well as random networks with asymptotic scale-free connectivity distributions characterized by an arbitrary scaling exponent α > 2. Our resul...

here, a predator-prey model with hassell-varley type functional responses is studied. some sufficient conditions are obtained for the permanence and global asymptotic stability of the system by using comparison theorem and constructing a suitable lyapunov functional. moreover, an example is illustrated to verify the results by simulation.