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

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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.

Endogeneizing know-how flows through the nature of R&D investments

In this paper we carefully link knowledge flows to and from a firms innovation process with this firms investment decisions. Three types of investments are considered: investments in applied research, investments in basic research, and investments in intellectual property protection. Only when basic research is performed, can the firm effectively access incoming knowledge flows and these incoming spillovers serve to increase the efficiency of own applied research.. The firm can at the same time influence outgoing knowledge flows, improving appropriability of its innovations, by investing in protection. Our results indicate that firms with small budgets for innovation will not invest in basic research. This occurs in the short run, when the budget for know-how creation is restricted, or in the long-run, when market opportunities are low, when legal protection is not very important, or, when the pool of accessible and relevant external know-how is limited. The ratio! of basic to applied research is non-decreasing in the size of the pool of accessible external know-how, the size and opportunity of the market, and, the effectiveness of intellectual property rights protection. This indicates the existence of economies of scale in basic research due to external market related factors. Empirical evidence from a sample of innovative manufacturing firms in Belgium confirms the economies of scale in basic research as a consequence of the firms capacity to access external knowledge flows and to protect intellectual property, as well as the complementarity between legal and strategic investments.

Pure braid subgroups of braided Thompson's groups

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Generalized property R and the Schoenflies Conjecture

There is a relation between the generalized Property R Conjecture and the Schoenflies Conjecture that suggests a new line of attack on the latter. The new approach gives a quick proof of the genus 2 Schoenflies Conjecture and suffices to prove the genus 3 case, even in the absence of new progress on the generalized Property R Conjecture.

Firm strategies in R&D: cooperation and participation in R&D programs

This paper presents empirical evidence on firms’ decisions to cooperate in R&D and the extent to which government sponsored R&D programs increase cooperation. Using a sample of firms from the Spanish innovation survey we jointly estimate the determinants of firm participation in R&D programs and of choice of cooperation partners. We find that (i) firms participating in national and European programs have different profiles, suggesting program complementarity; (ii) private-private and publicprivate cooperation are associated on average with firms with different characteristics, and (iii) national R&D programs seem to have a positive effect on all types of cooperation, but especially on public-private partnerships.

Who participates in R&D subsidy programs? The case of spanish manufacturing firms.

Empirical evidence on the effectiveness of R&D subsidies to firms has produced mixed results so far. One possible explanation is that firms and project selection rules may be quite heterogeneous both across agencies and across industries, leading to different outcomes in terms of the induced additional private effort. Here we focus on the participation stage. Using a sample of Spanish firms, we test for differences across agencies and industries. Our results suggest that firms in the same industry face different hurdles to participate in different agencies’ programs, that participation patterns may reflect a combination of agency goals, and that patterns differ across high-tech and low-tech industries.

Minimal lenght elements of Thompson's groups F(p)

We describe a method for determining the minimal length of elements in the generalized Thompson's groups F(p). We compute the length of an element by constructing a tree pair diagram for the element, classifying the nodes of the tree and summing associated weights from the pairs of node classifications. We use this method to effectively find minimal length representatives of an element.

Metric properties of the braided Thompson's groups

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Tree automorphisms and quasi-isometries of Thompson's group F

We prove that automorphisms of the infinite binary rooted tree T2 do not yield quasi-isometries of Thompson's group F, except for the map which reverses orientation on the unit interval, a natural outer automorphism of F. This map, together with the identity map, forms a subgroup of Aut(T2) consisting of 2-adic automorphisms, following standard terminology used in the study of branch groups. However, for more general p, we show that the analgous groups of p-adic tree automorphisms do not give rise to quasiisometries of F(p).

Assessing the assignation of public subsidies: Do the experts choose the most efficient R&D projects?

The implementation of public programs to support business R&D projects requires the establishment of a selection process. This selection process faces various difficulties, which include the measurement of the impact of the R&D projects as well as selection process optimization among projects with multiple, and sometimes incomparable, performance indicators. To this end, public agencies generally use the peer review method, which, while presenting some advantages, also demonstrates significant drawbacks. Private firms, on the other hand, tend toward more quantitative methods, such as Data Envelopment Analysis (DEA), in their pursuit of R&D investment optimization. In this paper, the performance of a public agency peer review method of project selection is compared with an alternative DEA method.