On the mathematics of data centre network topologies.

In a recent paper, combinatorial designs were used to construct switch-centric data centre networks that compare favourably with the ubiquitous (enhanced) fat-tree data centre networks in terms of the number of servers within (given a fixed server-to-server diameter). Unfortunately there were flaws in some of the proofs in that paper. We correct these flaws here and extend the results so as to prove that the core combinatorial construction, namely the 3-step construction, results in data centre networks with optimal path diversity

On the mathematics of data centre network topologies

In a recent paper, combinatorial designs were used to construct switch-centric data centre networks that compare favourably with the ubiquitous (enhanced) fat-tree data centre networks in terms of the number of servers within (given a fixed server-to-server diameter). Unfortunately there were flaws in some of the proofs in that paper. We correct these flaws here and extend the results so as to prove that the core combinatorial construction, namely the 3-step construction, results in data centre networks with optimal path diversity

Minimal disconnected cuts in planar graphs

The problem of finding a disconnected cut in a graph is NP-hard in general but polynomial-time solvable on planar graphs. The problem of finding a minimal disconnected cut is also NP-hard but its computational complexity is not known for planar graphs. We show that it is polynomial-time solvable on 3-connected planar graphs but NP-hard for 2-connected planar graphs. Our technique for the first result is based on a structural characterization of minimal disconnected cuts in 3-connected K 3,3 -free-minor graphs and on solving a topological minor problem in the dual. We show that the latter problem can be solved in polynomial-time even on general graphs. In addition we show that the problem of finding a minimal connected cut of size at least 3 is NP-hard for 2-connected apex graphs

When Patrolmen Become Corrupted: Monitoring a Graph Using Faulty Mobile Robots

A team of k mobile robots is deployed on a weighted graph whose edge weights represent distances. The robots move perpetually along the domain, represented by all points belonging to the graph edges, without exceeding their maximum speed. The robots need to patrol the graph by regularly visiting all points of the domain. In this paper, we consider a team of robots (patrolmen), at most f of which may be unreliable, i.e., they fail to comply with their patrolling duties. What algorithm should be followed so as to minimize the maximum time between successive visits of every edge point by a reliable patrolman? The corresponding measure of efficiency of patrolling called idleness has been widely accepted in the robotics literature. We extend it to the case of untrusted patrolmen; we denote by Ifk(G) the maximum time that a point of the domain may remain unvisited by reliable patrolmen. The objective is to find patrolling strategies minimizing Ifk(G). We investigate this problem for various classes of graphs. We design optimal algorithms for line segments, which turn out to be surprisingly different from strategies for related patrolling problems proposed in the literature. We then use these results to study general graphs. For Eulerian graphs G, we give an optimal patrolling strategy with idleness Ifk(G)=(f+1)|E|/k, where |E| is the sum of the lengths of the edges of G. Further, we show the hardness of the problem of computing the idle time for three robots, at most one of which is faulty, by reduction from 3-edge-coloring of cubic graphs—a known NP-hard problem. A byproduct of our proof is the investigation of classes of graphs minimizing idle time (with respect to the total length of edges); an example of such a class is known in the literature under the name of Kotzig graphs

Multiple random walks on paths and grids

We derive several new results on multiple random walks on "low dimensional" graphs. First, inspired by an example of a weighted random walk on a path of three vertices given by Efremenko and Reingold, we prove the following dichotomy: as the path length n tends to infinity, we have a super-linear speed-up w.r.t. the cover time if and only if the number of walks k is equal to 2. An important ingredient of our proofs is the use of a continuous-time analogue of multiple random walks, which might be of independent interest. Finally, we also present the first tight bounds on the speed-up of the cover time for any d-dimensional grid with d >= 2 being an arbitrary constant, and reveal a sharp transition between linear and logarithmic speed-up

Could an analysis of mean corpuscular volume help to improve risk stratification in non-anemic patients with acute myocardial infarction?

Background: Nowadays, when the majority of patients with acute myocardial infarction (AMI) are treated with primary percutaneous coronary intervention and modern pharmacotherapy, risk stratification becomes a challenge. Simple and easily accessible parameters that would help in a better determination of prognosis are needed. The aim of the study was to estimate the prevalence of high mean corpuscular volume (MCV, defined as MCV > 92 fL) and to establish its prognostic value in non-anemic patients with AMI. Methods: We retrospectively analyzed the data of 248 consecutive non-anemic patients hospitalized due to AMI (median age: 65 [59–76] years, men: 63%, ST segment elevation myocardial infarction: 31%, and median left ventricular ejection fraction [LVEF]: 50%). Results: The prevalence of high MCV was 39 ± 6% (± 95% confidence interval) in the entire AMI population. High MCV was more prevalent in males, patients with low body mass index, non-diabetics and cigarette smokers (all p < 0.05). During the 180-day follow-up, there were 38 (15%) events, defined as another AMI or death. In a multivariable Cox proportional hazard model, female gender (p < 0.01), low LVEF (p < 0.001), previous AMI (p < 0.05), arterial hypertension (p < 0.05), and high MCV (p < 0.001) were prognosticators of pre-defined events. Conclusions: In non-anemic patients with AMI, high MCV is an independent prognostic factor of poor outcome defined as another AMI or death.

Strategies used as spectroscopy of financial markets reveal new stylized facts

We propose a new set of stylized facts quantifying the structure of financial markets. The key idea is to study the combined structure of both investment strategies and prices in order to open a qualitatively new level of understanding of financial and economic markets. We study the detailed order flow on the Shenzhen Stock Exchange of China for the whole year of 2003. This enormous dataset allows us to compare (i) a closed national market (A-shares) with an international market (B-shares), (ii) individuals and institutions and (iii) real investors to random strategies with respect to timing that share otherwise all other characteristics. We find that more trading results in smaller net return due to trading frictions. We unveiled quantitative power laws with non-trivial exponents, that quantify the deterioration of performance with frequency and with holding period of the strategies used by investors. Random strategies are found to perform much better than real ones, both for winners and losers. Surprising large arbitrage opportunities exist, especially when using zero-intelligence strategies. This is a diagnostic of possible inefficiencies of these financial markets.Comment: 13 pages including 5 figures and 1 tabl

On convergence and threshold properties of discrete lotka-volterra population protocols

In this work we focus on a natural class of population protocols whose dynamics are modeled by the discrete version of Lotka-Volterra equations with no linear term. In such protocols, when an agent a of type (species) i interacts with an agent b of type (species) j with a as the initiator, then b’s type becomes i with probability Pij. In such an interaction, we think of a as the predator, b as the prey, and the type of the prey is either converted to that of the predator or stays as is. Such protocols capture the dynamics of some opinion spreading models and generalize the well-known Rock-Paper-Scissors discrete dynamics. We consider the pairwise interactions among agents that are scheduled uniformly at random. We start by considering the convergence time and show that any Lotka-Volterra-type protocol on an n-agent populati

The Minangkabau Healers And Healing Methods: A Structural Analysis

In this thesis the researcher wants to illustrate the healing methods of the Minangkabau in West Sumatra. It looks at one village of the interior and describes the different healing methods of the Minangkabau in that village. The aim is to illustrate the different ways of treatment and to analyze it in a structural way by following the ideas of Claude Levi-Strauss and Josselin de Jong. For many anthropologists, structuralism and its methodology are outdated, but this thesis intends to show that a structural approach is still fruitful and could contribute to analyze traditional healing methods. Furthermore, this thesis will illustrate that there are different types of healers with their own ways of treatment within one village society and its network. The research objectives are to explore the Minangkabau healers and their healing methods. The different types of healers, healing methods and plants should be categorized. This thesis will also examine whether Frederick Errington’s hypothesis that the Minangkabau are sign-oriented is correct in the field of healing. The Minangkabau society is both part of Southeast Asia and part of the Islamic world and therefore this research shows in how far these healing methods are embedded within a greater context. The research discovers that there are three types of healers who still play an important role in the health of the local population. There are three elements of healing methods that play a certain role. Unique patterns of the Minangkabau healing methods are described by the researcher and give an impression why traditional healing methods are still relevant

Almost optimal asynchronous rendezvous in infinite multidimensional grids

Two anonymous mobile agents (robots) moving in an asynchronous manner have to meet in an infinite grid of dimension δ> 0, starting from two arbitrary positions at distance at most d. Since the problem is clearly infeasible in such general setting, we assume that the grid is embedded in a δ-dimensional Euclidean space and that each agent knows the Cartesian coordinates of its own initial position (but not the one of the other agent). We design an algorithm permitting the agents to meet after traversing a trajectory of length O(d δ polylog d). This bound for the case of 2d-grids subsumes the main result of [12]. The algorithm is almost optimal, since the Ω(d δ) lower bound is straightforward. Further, we apply our rendezvous method to the following network design problem. The ports of the δ-dimensional grid have to be set such that two anonymous agents starting at distance at most d from each other will always meet, moving in an asynchronous manner, after traversing a O(d δ polylog d) length trajectory. We can also apply our method to a version of the geometric rendezvous problem. Two anonymous agents move asynchronously in the δ-dimensional Euclidean space. The agents have the radii of visibility of r1 and r2, respectively. Each agent knows only its own initial position and its own radius of visibility. The agents meet when one agent is visible to the other one. We propose an algorithm designing the trajectory of each agent, so that they always meet after traveling a total distance of O( ( d)), where r = min(r1, r2) and for r ≥ 1. r)δpolylog ( d r