On the topological classification of binary trees using the Horton-Strahler index

The Horton-Strahler (HS) index $r=\max{(i,j)}+\delta_{i,j}$ has been shown to be relevant to a number of physical (such at diffusion limited aggregation) geological (river networks), biological (pulmonary arteries, blood vessels, various species of trees) and computational (use of registers) applications. Here we revisit the enumeration problem of the HS index on the rooted, unlabeled, plane binary set of trees, and enumerate the same index on the ambilateral set of rooted, plane binary set of trees of $n$ leaves. The ambilateral set is a set of trees whose elements cannot be obtained from each other via an arbitrary number of reflections with respect to vertical axes passing through any of the nodes on the tree. For the unlabeled set we give an alternate derivation to the existing exact solution. Extending this technique for the ambilateral set, which is described by an infinite series of non-linear functional equations, we are able to give a double-exponentially converging approximant to the generating functions in a neighborhood of their convergence circle, and derive an explicit asymptotic form for the number of such trees.Comment: 14 pages, 7 embedded postscript figures, some minor changes and typos correcte

Non-crossing dependencies: Least effort, not grammar

The use of null hypotheses (in a statistical sense) is common in hard sciences but not in theoretical linguistics. Here the null hypothesis that the low frequency of syntactic dependency crossings is expected by an arbitrary ordering of words is rejected. It is shown that this would require star dependency structures, which are both unrealistic and too restrictive. The hypothesis of the limited resources of the human brain is revisited. Stronger null hypotheses taking into account actual dependency lengths for the likelihood of crossings are presented. Those hypotheses suggests that crossings are likely to reduce when dependencies are shortened. A hypothesis based on pressure to reduce dependency lengths is more parsimonious than a principle of minimization of crossings or a grammatical ban that is totally dissociated from the general and non-linguistic principle of economy.Postprint (author's final draft

Induced cycle structures of the hyperoctahedral group

The hyperoctahedral group B{sub n} is treated as the automorphism group of the n-dimensional hypercube, denoted Q{sub n}, which is nowadays understood to be a graph on 2{sup n} vertices. It is well-known that B{sub n} can be represented by the group of signed permutations. In other words, any signed permutation induces a permutation on the vertices of Q{sub n} which preserves adjacencies. Moreover, signed permutations also a permutation group on the edge of Q{sub n}, denoted H{sub n}. We study the cycle structures of both B{sub n} and H{sub n}. The technique proposed here is to determine the induced cycle structure of a signed permutation by the number of fixed vertices or fixed edges of a signed permutation in the cyclic group generated by a signed permutation of given type. Here we directly define the type of a signed permutation by a double partition based on its signed cycle decomposition. In this way, we obtain explicit formulas for the number of induced cycles on vertices as well as edges of Q{sub n} of a signed permutation in terms of its type. By further exploring the connection between cycle indices and the structure of fixed points, we obtain the cycle indices of both B{sub n} and H{sub n}. Our formula for the cycle index of B{sub n}is much more natural and considerably simpler than that of Harrison and High. Meanwhile, the cycle structure of H{sub n} seems to have been untouched before, although it is well motivated by nonisomorphic edge colorings of Q{sub n} as well as by the recent interest in symmetries of computer networks

What indicators matter? The Analysis of Perception toward Research Assessment Indicators and Leiden Manifesto- The Case Study of Taiwan

This study aims to investigate the Taiwanese researchers' awareness toward bibliometric indicators and the principles from Leiden Manifesto. The online survey was conducted and obtained a total of 417 valid responses. The results show that evoking the right concept of use of bibliometric indicators and research evaluation has a long way to go. The lack of recognition of bibliometric indicators exists in Taiwan academia. Generally speaking, researchers may hear of the certain indicators, but they are not familiar its definition and calculation process. Only JIF and h-index are considered as well-known indicators. Compared to San Francisco Declaration on Research Assessment (DORA), Leiden Manifesto has lower visibility in Taiwan, hence, this study also aims to promote the right concepts of research evaluation by conducting questionnaire survey. The analysis results suggest that the ten principles can be considered the universal guideline in research evaluation since most of Taiwanese researchers agree the contents of ten principles. Especially for the principle 6 “Account for variation by field in publication and citation practices” has less room of opinion diversity. However, it is interesting to compare the result of recognition of relative citation ratio, only few researchers have fully understood the definition. This result indicates that scientometricians should need to make more effort to disseminate the concept of field-normalization in bibliometric indicators. The researchers do have understanding about the importance of comparison on the same basis, at the meantime, they may use the inappropriate indicators just because lack of enough knowledge on the variety of indicators. Hence, it is important to initiate that education of informetrics to all of the stakeholders in research evaluation so that the misuse and abuse of bibliometric indicators may possibly do not happen again, and the bibliometric analysis is able to turn to contextualization-based analysis in the future

What indicators matter? The Analysis of Perception toward Research Assessment Indicators and Leiden Manifesto- The Case Study of Taiwan

This study aims to investigate the Taiwanese researchers' awareness toward bibliometric indicators and the principles from Leiden Manifesto. The online survey was conducted and obtained a total of 417 valid responses. The results show that evoking the right concept of use of bibliometric indicators and research evaluation has a long way to go. The lack of recognition of bibliometric indicators exists in Taiwan academia. Generally speaking, researchers may hear of the certain indicators, but they are not familiar its definition and calculation process. Only JIF and h-index are considered as well-known indicators. Compared to San Francisco Declaration on Research Assessment (DORA), Leiden Manifesto has lower visibility in Taiwan, hence, this study also aims to promote the right concepts of research evaluation by conducting questionnaire survey. The analysis results suggest that the ten principles can be considered the universal guideline in research evaluation since most of Taiwanese researchers agree the contents of ten principles. Especially for the principle 6 “Account for variation by field in publication and citation practices” has less room of opinion diversity. However, it is interesting to compare the result of recognition of relative citation ratio, only few researchers have fully understood the definition. This result indicates that scientometricians should need to make more effort to disseminate the concept of field-normalization in bibliometric indicators. The researchers do have understanding about the importance of comparison on the same basis, at the meantime, they may use the inappropriate indicators just because lack of enough knowledge on the variety of indicators. Hence, it is important to initiate that education of informetrics to all of the stakeholders in research evaluation so that the misuse and abuse of bibliometric indicators may possibly do not happen again, and the bibliometric analysis is able to turn to contextualization-based analysis in the future

Combinatorics, geometry, and mathematical physics

This is the final report of a three-year, Laboratory-Directed Research and Development (LDRD) project at the Los Alamos National Laboratory (LANL). Combinatorics and geometry have been among the most active areas of mathematics over the past few years because of newly discovered inter-relations between them and their potential for applications. In this project, the authors set out to identify problems in physics, chemistry, and biology where these methods could impact significantly. In particular, the experience suggested that the areas of unitary symmetry and discrete dynamical systems could be brought more strongly under the purview of combinatorial methods. Unitary symmetry deals with the detailed description of the quantum mechanics of many-particle systems, and discrete dynamical systems with chaotic systems. The depth and complexity of the mathematics in these physical areas of research suggested that not only could significant advances be made in these areas, but also that here would be a fertile feedback of concept and structure to enrich combinatorics itself by setting new directions. During the three years of this project, the goals have been realized beyond expectation, and in this report the authors set forth these advancements and justify their optimism