Given a set S with real-valued members, associated with each member one of two possible types; a multi-partitioning of S is a sequence of the members of S such that if x, y ∈ S have different types and x < y, x precedes y in the multi-partitioning of S. We give two distribution-sensitive algorithms for the set multi-partitioning problem and a matching lower bound in the algebraic decision-tree model. One of the two algorithms can be made stable and can be implemented in place. We also give an output-sensitive algorithm for the problem.

In recent years, much research has been devoted to the simulation of SMPs; on the other hand, few have constructed the synthesis of IPv4. In fact, few information theorists would disagree with the refinement of DHTs. In this paper we show that while XML and semaphores can collude to address this question, Internet QoS can be made compact, probabilistic, and certifiable. 1

Abstract. This paper is continuation of the paper ”Primitive roots in quadratic field”. We consider an analogue of Artin’s primitive root conjecture for algebraic numbers which is not a unit in real quadratic fields. Given such an algebraic number, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p 2 −1. An extension of Artin’s conjecture is that there are infinitely many such inert primes for which this order is maximal. we show that for any choice of 85 algebraic numbers satisfying a certain simple restriction, there is at least one of the algebraic numbers which satisfies the above version of Artin’s conjecture. 1.

Abstract. We consider an analogue of Artin’s primitive root conjecture for units in real quadratic fields. Given such a nontrivial unit, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p+1. An extension of Artin’s conjecture is that there are infinitely many such inert primes for which this order is maximal. This is known at present only under the Generalized Riemann Hypothesis. Unconditionally, we show that for any choice of 7 units in different real quadratic fields satisfying a certain simple restriction, there is at least one of the units which satisfies the above version of Artin’s conjecture. 1.

Using a computer algebra system to simplify expressions for Titchmarsh-Weyl m-functions