Some common indexes of group diversity: upper boundaries

Workgroup diversity can be conceptualized as variety, separation, or disparity. Thus, the proper operationalization of diversity depends on how a diversity dimension has been defined. Analytically, the minimal diversity must be obtained when there are no differences on an attribute among the members of a group, however maximal diversity has a different shape for each conceptualization of diversity. Previous work on diversity indexes indicated maximum values for variety (e.g., Blau"s index and Teachman"s index), separation (e.g., standard deviation and mean Euclidean distance), and disparity (e.g., coefficient of variation and the Gini coefficient of concentration), although these maximum values are not valid for all group characteristics (i.e., group size and group size parity) and attribute scales (i.e., number of categories). We demonstrate analytically appropriate upper boundaries for conditional diversity determined by some specific group characteristics, avoiding the bias related to absolute diversity. This will allow applied researchers to make better interpretations regarding the relationship between group diversity and group outcomes.

Upper gastrointestinal bleeding in cirrhosis: clinical and endoscopic correlations

The clinical data of 180 episodes of upper gastrointestinal bleeding in 168 patients with cirrhosis of the liver are examined. The source of bleeding had been determined by early endoscopy in all cases. In men under the age of 50 years, and without symptoms of liver failure, bleeding was due to ruptured gastro-oesophageal varices in 84% of cases. Severe liver failure was associated with acute lesions of gastric mucosa in many cases. No presumptive diagnosis of the source of haemorrhage could be based on the examination of other clinical data (presence of ascites, mode of presentation and pattern of bleeding, history of ulcer disease, alcoholism, and previous medication.

Upper gastrointestinal bleeding in cirrhosis: clinical and endoscopic correlations

The clinical data of 180 episodes of upper gastrointestinal bleeding in 168 patients with cirrhosis of the liver are examined. The source of bleeding had been determined by early endoscopy in all cases. In men under the age of 50 years, and without symptoms of liver failure, bleeding was due to ruptured gastro-oesophageal varices in 84% of cases. Severe liver failure was associated with acute lesions of gastric mucosa in many cases. No presumptive diagnosis of the source of haemorrhage could be based on the examination of other clinical data (presence of ascites, mode of presentation and pattern of bleeding, history of ulcer disease, alcoholism, and previous medication.

Comparative study of upper lip frenectomy with the CO2 laser versus the Er, Cr: YSGG laser

Objectives: To compare upper lip frenulum reinsertion, bleeding, surgical time and surgical wound healing in frenectomies performed with the CO2 laser versus the Er, Cr:YSGG laser. Study design: A prospective study was carried out on 50 randomized pediatric patients who underwent rhomboidal resection of the upper lip frenulum with either the CO2 laser or the Er,Cr:YSGG laser. Twenty-five patients were assigned to each laser system. All patients were examined at 7, 14, 21 days and 4 months after the operation in order to assess the surgical wound healing. Results: Insertion of the frenulum, which was preoperatively located between the upper central incisors, migrated to the mucogingival junction as a result of using both laser systems in all patients. Only two patients required a single dose of 650 mg of paracetamol, one of either study group. CO2 laser registered improved intraoperative bleeding control results and shorter surgical times. On the other hand, the Er,Cr:YSGG laser achieved faster healing. Conclusions: Upper lip laser frenectomy is a simple technique that results in minimum or no postoperative swelling or pain, and which involves upper lip frenulum reinsertion at the mucogingival junction. The CO2 laser offers a bloodless field and shorter surgical times compared with the Er,Cr:YSGG laser. On the other hand, the Er,Cr:YSGG laser achieved faster wound healing.

The unbounded dead-end depth propertie is not a group invariant

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Examples of retracts in a free group that are not the fixed subgroup of any group of automorphisms

Let F be a free group of rank at least three. We show that some retracts of F previously studied by Martino-Ventura are not equal to the fixed subgroup of any group of automorphisms of F. This shows that, in F, there exist subgroups that are equal to the fixed subgroup of some set of endomorphisms but are not equal to the fixed subgroup of any set of automorphisms. Moreover, we determine the Galois monoids of these retracts, where, by the Galois monoid of a subgroup H of F, we mean the monoid consisting of all endomorphisms of F that fix H.

The automorphism group of a free-by-cyclic group in rank 2

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On an upper bound for the arithmetic self-intersection number of the dualizing sheaf on arithmetic surfaces

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Growth of positive words and lower bounds of the growth rate for Thompson's group (F)p

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Bounding right-arm rotation distances

Rotation distance quantifies the difference in shape between two rooted binary trees of the same size by counting the minimum number of elementary changes needed to transform one tree to the other. We describe several types of rotation distance, and provide upper bounds on distances between trees with a fixed number of nodes with respect to each type. These bounds are obtained by relating each restricted rotation distance to the word length of elements of Thompson's group F with respect to different generating sets, including both finite and infinite generating sets.

Computational explorations in Thompson's group F

Here we describe the results of some computational explorations in Thompson's group F. We describe experiments to estimate the cogrowth of F with respect to its standard finite generating set, designed to address the subtle and difficult question whether or not Thompson's group is amenable. We also describe experiments to estimate the exponential growth rate of F and the rate of escape of symmetric random walks with respect to the standard generating set.

Combinatorial and metric properties of Thompson's group T

We discuss metric and combinatorial properties of Thompson's group T, such as the normal forms for elements and uniqueness of tree pair diagrams. We relate these properties to those of Thompson's group F when possible, and highlight combinatorial differences between the two groups. We define a set of unique normal forms for elements of T arising from minimal factorizations of elements into convenient pieces. We show that the number of carets in a reduced representative of T estimates the word length, that F is undistorted in T, and that cyclic subgroups of T are undistorted. We show that every element of T has a power which is conjugate to an element of F and describe how to recognize torsion elements in T.

Testing Cayley graph densities

We present a computer-assisted analysis of combinatorial properties of the Cayley graphs of certain finitely generated groups: Given a group with a finite set of generators, we study the density of the corresponding Cayley graph, that is, the least upper bound for the average vertex degree (= number of adjacent edges) of any finite subgraph. It is known that an m-generated group is amenable if and only if the density of the corresponding Cayley graph equals to 2m. We test amenable and non-amenable groups, and also groups for which amenability is unknown. In the latter class we focus on Richard Thompson’s group F.

On group strategy-proof mechanisms for a many-to-one matching model

For the many-to-one matching model in which firms have substitutable and quota q-separable preferences over subsets of workers we show that the workers-optimal stable mechanism is group strategy-proof for the workers. In order to prove this result, we also show that under this domain of preferences (which contains the domain of responsive preferences of the college admissions problem) the workers-optimal stable matching is weakly Pareto optimal for the workers and the Blocking Lemma holds as well. We exhibit an example showing that none of these three results remain true if the preferences of firms are substitutable but not quota q-separable.

When does universal peace prevail? Secession and group formation in rent seeking contests and policy conflicts

This paper analyzes secession and group formation in a general model of contest inspired by Esteban and Ray (1999). This model encompasses as special cases rent seeking contests and policy conflicts, where agents lobby over the choice of a policy in a one-dimensional policy space. We show that in both models the grand coalition is the efficient coalition structure and agents are always better off in the grand coalition than in a symmetric coalition structure. Individual agents (in the rent seeking contest) and extremists (in the policy conflict) only have an incentive to secede when they anticipate that their secession will not be followed by additional secessions. Incentives to secede are lower when agents cooperate inside groups. The grand coalition emerges as the unique subgame perfect equilibrium outcome of a sequential game of coalition formation in rent seeking contests.