Topological Event Research Articles

A projective formalism applied to topological and probabilistic event structures

This paper introduces projective systems for topological and probabilistic event structures. The projective formalism is used for studying the domain of configurations of a prime event structure and its space of maximal elements. This is done from both a topological and a probabilistic viewpoint. We give probability measure extension theorems in this framework.

General compact labeling schemes for dynamic trees

Let F be a function on pairs of vertices. An F-labeling scheme is composed of a marker algorithm for labeling the vertices of a graph with short labels, coupled with a decoder algorithm allowing one to compute F(u, v) for any two vertices u and v directly from their labels. As applications for labeling schemes concern mainly large and dynamically changing networks, it is of interest to study distributed dynamic labeling schemes. This paper investigates labeling schemes for dynamic trees. We consider two dynamic tree models, namely, the leaf-dynamic tree model in which at each step a leaf can be added to or removed from the tree and the leaf-increasing tree model in which the only topological event that may occur is that a leaf joins the tree. A general method for constructing labeling schemes for dynamic trees (under the above mentioned dynamic tree models) was previously developed in Korman et al. (Theory Comput Syst 37:49–75, 2004). This method is based on extending an existing static tree labeling scheme to the dynamic setting. This approach fits many natural functions on trees, such as distance, separation level, ancestry relation, routing (in both the adversary and the designer port models), nearest common ancestor etc.. Their resulting dynamic schemes incur overheads (over the static scheme) on the label size and on the communication complexity. In particular, all their schemes yield a multiplicative overhead factor of Ω(log n) on the label sizes of the static schemes. Following (Korman et al., Theory Comput Syst 37:49–75, 2004), we develop a different general method for extending static labeling schemes to the dynamic tree settings. Our method fits the same class of tree functions. In contrast to the above paper, our trade-off is designed to minimize the label size, sometimes at the expense of communication. Informally, for any function k(n) and any static F-labeling scheme on trees, we present an F-labeling scheme on dynamic trees incurring multiplicative overhead factors (over the static scheme) of $$O(\log_{k(n)} n)$$ on the label size and $$O(k(n)\log_{k(n)} n)$$ on the amortized message complexity. In particular, by setting $$k(n) = n^{\epsilon}$$ for any $$0 < \epsilon < 1$$ , we obtain dynamic labeling schemes with asymptotically optimal label sizes and sublinear amortized message complexity for the ancestry relation, the id-based and label-based nearest common ancestor relation and the routing function.

A Two-Dimensional Kinetic Triangulation with Near-Quadratic Topological Changes

A triangulation of a set S of points in the plane is a subdivision of the convex hull of S into triangles whose vertices are points of S. Given a set S of n points in \({\Bbb R}^2,\) each moving independently, we wish to maintain a triangulation of S. The triangulation needs to be updated periodically as the points in S move, so the goal is to maintain a triangulation with a small number of topological events, each being the insertion or deletion of an edge. We propose a kinetic data structure (KDS) that processes \(n^2 2^{O(\sqrt{\log n \cdot \log \log n})}\) topological events with high probability if the trajectories of input points are algebraic curves of fixed degree. Each topological event can be processed in \(O(\log n)\) time. This is the first known KDS for maintaining a triangulation that processes a near-quadratic number of topological events, and almost matches the \(\Omega(n^2)\) lower bound [1]. The number of topological events can be reduced to \(nk\cdot 2^{O(\sqrt{\log k \cdot \log\log n})}\) if only k of the points are moving.

Updating the topology of the dynamic Voronoi diagram for spheres in Euclidean d-dimensional space

The problem of dynamic maintenance of a Voronoi diagram for a set of spheres moving independently in d-dimensional space is addressed in this paper. The maintenance of this Voronoi diagram for spheres moving along given trajectories, requires the calculation of topological events, that occur when d+2 spheres become tangent to a common sphere. The criterion for determination of the topological event in the Euclidean metric is derived as a solution of a system of non-linear algebraic equations. The criterion is given in the form of polynomial algebraic equations dependent on the coordinates and trajectories of the moving spheres. These equations are solved using numerical methods. Application of the method to study the structure of a system of polydisperse spheres in a three-dimensional Euclidean space is briefly discussed.

Implementation and synchronisation of the First Level Global Trigger for the CMS experiment at LHC

The hardware implementation of the First Level Global Trigger for the Compact Muon Solenoid experiment at the CERN Large Hadron Collider is described. Special emphasis is given to the algorithm logic and the synchronisation procedure. Up to 128 different trigger algorithms are calculated in parallel by the Global Trigger (GT) for every beam crossing taking place at 25 ns intervals. Already, at the first trigger level the GT is able to select complex topological event configurations by performing fast calculations. The electronics is based on VME and relies completely on Field Programmable Gate Arrays (FPGA) technology. The electronic circuits are optimised for speed by exploiting, to a great extent, the small look-up tables provided in the FPGA chips.

Concept, design and performance of the second level triggers of the H1 detector

Since the end of the 1995 HERA running period the second level (L2) trigger system of the H1 experiment has been in operation. The L2 decision is presently derived from a combination of two independent hardware systems; one is based on the output of neural networks while the other decides on the grounds of topological event signatures in the detector. In contrast to the first level trigger (L1), which is fully pipelined and operates at the HERA bunch crossing rate of 10.4 MHz, the decision of the L2 systems is delivered within 20 /spl mu/s of the preceding L1 keep signal, allowing sufficient time for transmission of trigger information and computation of the L2 trigger response. This scheme significantly reduces the trigger rate, with minimal impact on detector deadtime for L1 rates below 1 kHz. The L2 trigger has proven to be flexible and highly effective in isolating physics processes from background.

QCD background for the UA1 W-->tb-bar signal.

The cc-barX and bb-barX background affecting the W..-->..tb-bar e/..mu..+2 jets signal reported by the UA1 Collaboration is discussed. Calculations are based on an O(..cap alpha../sub s/ /sup 2/)+leading-pole approximation scheme, which includes jet radiation in the initial and final states of the hard binary parton process, and include the heavy-flavor excitation contribution, the stability of which has been checked. The main conclusions are the following. (i) Selection cuts naturally induce a topological event structure on the surviving background mimicking that expected for a W..-->..tb-bar signal. (ii) The background rates for ..mu..+2 jets events, on which we concentrate, are compatible within theoretical uncertainties with the number of events experimentally observed. The importance of purely leptonic decay modes, which survive the lepton-isolation cuts, and the bias induced by lepton-isolation cuts on background event topologies are also clarified.