In this paper we construct the maximal subgroups of geometric type of the orthogonal groups in dimension d over GF(q) in O(d3+d2log q+log qlog log q) finite field operations.

Since any function f(x1, … , xm) from {0, 1}m in a finite field k can be uniquely written as a multilinear polynomial, we associate to it its inverse dual f*(x1, … , xm) expressing the coefficients of this canonical polynomial. We show that the unlikely hypothesis that the class P(k) of sequences of functions of polynomial complexity be closed by duality is equivalent to the well-known hypothesis P = #pP, where p is the characteristic of k.

In a first section we expose the result in the frame of classical Boolean calculus, that is, when k = ℤ/2ℤ. In a second section we treat the general case, introducing a notion of transformation whose duality is a special case; the transformations form a group isomorphic to GL2(k); among them, we distinguish the benign transformations, which have a weak effect on the complexity of functions; we show that, in this respect, all the non-benign transformations have the same power of harmfulness.

In the third section we consider functions from km into k, and in the last, after introducing #P = P to the landscape, we compare our results with those of Guillaume Malod, concerning the closure by ‘coefficient-function’ of various classes of complexity of sequences of polynomial defined in Valiant's way.

After suitable normalization the asymptotic root value W of a minimax game tree of order b ≥ 2 with independent and identically distributed input values having a continuous, strictly increasing distribution function on a subinterval of R appears to be a particular solution of the stochastic maximin fixed-point equation W ξ max1≤i≤bmin1≤j≤bWi,j, where Wi,j are independent copies of W and denotes equality in law. Moreover, ξ= g'(α) > 1, where g(x) := (1 − (1 − x)b)b and α denotes the unique fixed point of g in (0, 1). This equation, which takes the form F(t) = g(F(t/ξ)) in terms of the distribution function F of W, is studied in the present paper for a reasonably extended class of functions g so as to encompass more general stochastic maximin equations as well. A complete description of the set of solutions F is provided followed by a discussion of additional properties such as continuity, differentiability, or existence of moments. Based on these results, it is further shown that the particular solution mentioned above stands out among all other ones in that its distribution function is the restriction of an entire function to the real line. This extends recent work of Ali Khan, Devroye and Neininger (2005). A connection with another class of stochastic fixed-point equations for weighted minima and maxima is also discussed.

We discuss two Monte Carlo algorithms for finding the global maximum of a simple random walk with negative drift. This problem can be used to connect the analysis of random input Monte Carlo algorithms with ideas and principles from mathematical statistics.

The iterative division of a triangle by chords which join a randomly-selected vertex of a triangle to the opposite side is investigated. Results on the limiting random graph which eventuates are given. Aspects studied are: the order of vertices; the fragmentation of chords; age distributions for elements of the graph; various topological characterisations of the triangles. Different sampling protocols are explored. Extensive use is made of the theory of branching processes.

This paper studies path lengths in random binary search trees under the random permutation model. It is known that the total path length, when properly normalized, converges almost surely to a nondegenerate random variable Z. The limit distribution is commonly referred to as the ‘quicksort distribution’. For the class 𝒜m of finite binary trees with at most m nodes we partition the external nodes of the binary search tree according to the largest tree that each external node belongs to. Thus, the external path length is divided into parts, each part associated with a tree in 𝒜m. We show that the vector of these path lengths, after normalization, converges almost surely to a constant vector times Z.

This paper is a study of the error in approximating the global maximum of a Brownian motion on the unit interval by observing the value at randomly chosen points. One point of view is to look at the error from random sampling for a given fixed Brownian sample path; another is to look at the error with both the path and observations random. In the first case we show that for almost all Brownian paths the error, normalized by multiplying by the square root of the number of observations, does not converge in distribution, while in the second case the normalized error does converge in distribution. We derive the limiting distribution of the normalized error averaged over all paths.

The transition probabilities of the move-to-front scheme are obtained by exploiting the connection between it and the coupon-collector's problem.

In this paper we study the following general class of concurrent processing systems. There are several different classes of processors (servers) and many identical processors within each class. There is also a continuous random flow of jobs, arriving for processing at the system. Each job needs to engage concurrently several processors from various classes in order to be processed. After acquiring the needed processors the job begins to be executed. Processing is done non-preemptively, lasts for a random amount of time, and then all the processors are released simultaneously. Each job is specified by its arrival time, its processing time, and the list of processors that it needs to access simultaneously. The random flow (sequence) of jobs has a stationary and ergodic structure. There are several possible policies for scheduling the jobs on the processors for execution; it is up to the system designer to choose the scheduling policy to achieve certain objectives.

We focus on the effect that the choice of scheduling policy has on the asymptotic behavior of the system at large times and especially on its stability, under general stationary and ergocic input flows.