Let Qn be the random number of comparisons made by quicksort in sorting n distinct keys, when we assume that all n! possible orderings are equally likely. Known results concerning moments for Qn do not show how rare it is for Qn to make large deviations from its mean. Here we give a good approximation to the probability of such a large deviation, and find that this probability is quite small. A...

We say that A ≤LR B if every B-random number is Arandom. Intuitively this means that if oracle A can identify some patterns on some real γ, oracle B can also find patterns on γ. In other words, B is at least as good as A for this purpose. We propose a methodology for studying the LR degrees and present a number of recent results of ours, including sketches of their proofs.

Let i, n be elements of N. The functor proj(i, n) yielding a function from Rn into R is defined by: (Def. 1) For every element x of Rn holds (proj(i, n))(x) = x(i). Next we state two propositions: (1) dom proj(1, 1) = R1 and rng proj(1, 1) = R and for every element x of R holds (proj(1, 1))(〈x〉) = x and (proj(1, 1))−1(x) = 〈x〉. (2)(i) (proj(1, 1))−1 is a function from R into R1, (ii) (proj(1, 1...

Using a new method to extract the data from various one-dimensional chaotic maps, we show that there is a nice correlation between the sign of the Lyapunov exponent of the maps and whether the extracted data form a good set of pseudo-random numbers using various well-known criteria.  2004 Elsevier B.V. All rights reserved. PACS: 05.45.+b; 05.40.+j; 02.50.sk; 02.60.-x

We consider a generalized form of the coupon collection problem in which a random number, S, of balls is drawn at each stage from an urn initially containing n white balls (coupons). Each white ball drawn is colored red and returned to the urn; red balls drawn are simply returned to the urn. The question considered is then: how many white balls (uncollected coupons) remain in the urn after the ...

By means of a simple OMNeT++ simulation scenario, which uses parallel streams of random numbers, we want to point out several technical pitfalls when applying different random number generators and using different initialization values for them. We describe shortcomings of the built-in random number generator (RNG) and describe traps when using modern RNGs. Quantitative and qualitative analyses...

In this paper, we apply evolutionary games to non-cooperative power control in wireless networks. Specifically, we focus our study in a power control in W-CDMA and WIMAX wireless systems. We study competitive power control within a large population of mobiles that interfere with each other through many local interactions. Each local interaction involves a random number of mobiles. An utility fu...

In this paper Y denotes a non empty set and G denotes a subset of PARTITIONS(Y ). Let X be a set. Then PARTITIONS(X) is a partition family of X. Let X be a set and let F be a non empty partition family of X. We see that the element of F is a partition of X. The following proposition is true (1) Let y be an element of Y . Then there exists a subset X of Y such that (i) y ∈ X, and (ii) there exis...

In this paper we design a protocol to extract random bits with an arbitrarily low bias from a single arbitrarily weak min-entropy block source in a device independent setting. The protocol employs Mermin devices that exhibit super-classical correlations. Number of devices used scales polynomially in the length of the block n, containing entropy of at least two bits. Our protocol is robust, it c...

For simplicity we adopt the following rules: x is arbitrary, m, n are natural numbers, f , g are functions, and A, B are sets. We now state several propositions: (1) For every function f and for every set X such that rng f ⊆ X holds idX · f = f. (2) Let X be a set, and let Y be a non empty set, and let f be a function from X into Y . Suppose f is one-to-one. Let B be a subset of X and let C be ...