How to Improve the Reliability of Chord?
In this paper we focus on Chord P2P protocol and we study the process of unexpected departures of nodes from this system. Each of such departures may effect in losing any information and in classical versions of this protocol the probability of losing some information is proportional to the quantity of information put into this system. This effect can be partially solved by gathering in the protocol multiple copies (replicas) of information. The replication mechanism was proposed by many authors. We present a detailed analysis of one variant of blind replication and show that this solution only partially solves the problem. Next we propose two less obvious modifications of the Chord protoco…
A Note on Algebraic Sums of Subsets of the Real Line
AbstractWe investigate the algebraic sums of sets for a large class of invari-ant ˙-ideals and ˙- elds of subsets of the real line. We give a simpleexample of two Borel subsets of the real line such that its algebraicsum is not a Borel set. Next we show a similar result to Proposition 2from A. Kharazishvili paper [4]. Our results are obtained for ideals withcoanalytical bases. 1 Introduction We shall work in ZFC set theory. By !we denote natural numbers. By 4wedenote the symmetric di erence of sets. The cardinality of a set Xwe denoteby jXj. By R we denote the real line and by Q we denote rational numbers. IfAand Bare subsets of R n and b2R , then A+B= fa+b: a2A^b2Bgand A+ b= A+ fbg. Simila…
Minimal Büchi Automata for Certain Classes of LTL Formulas
In this paper we calculate the minimal number of states of Buchi automata which encode some classes of linear temporal logic (LTL) formulas that are frequently used in model checking. Our results may be used for verification of the quality of algorithms which automatically translate LTL formulas into Buchi automata and for improving the quality and speed of such translators. In the last section of this paper we compare our lower-bound estimations to Buchi automata generated by two currently used translators: LTL2BA and SPOT.
Hamel-isomorphic images of the unit ball
In this article we consider linear isomorphisms over the field of rational numbers between the linear spaces ℝ2 and ℝ. We prove that if f is such an isomorphism, then the image by f of the unit disk is a strictly nonmeasurable subset of the real line, which has different properties than classical non-measurable subsets of reals. We shall also consider the question whether all images of bounded measurable subsets of the plane via a such mapping are non-measurable (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)