General primitivity in the mapping class group

For $g\geq 2$, let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g$. In this paper, we obtain necessary and sufficient conditions under which a given pseudo-periodic mapping class can be a root of another up to conjugacy. Using this characterization, the canonical decomposition of (non-periodic) mapping classes, and some known algorithms, we give an efficient algorithm for computing roots of arbitrary mapping classes up to conjugacy. Furthermore, we derive realizable bounds on the degrees of roots of pseudo-periodic mapping classes in $\mathrm{Mod}(S_g)$, the Torelli group, the level-$m$ subgroup of $\mathrm{Mod}(S_g)$, and the commutator subgroup of $\mathrm{Mod}(S_2)$. In particular, we show that the highest possible (realizable) degree of a root of a pseudo-periodic mapping class $F$ is $3q(F)(g+1)(g+2)$, where $q(F)$ is a unique positive integer associated with the conjugacy class of $F$. Moreover, this bound is realized by the roots of the powers of Dehn twist about a separating curve of genus $[g/2]$ in $S_g$. Finally, for $g\geq 3$, we show that any pseudo-periodic mapping class having a nontrivial periodic component that is not the hyperelliptic involution, normally generates $\mathrm{Mod}(S_g)$. Consequently, we establish that there exist roots of bounding pair maps and powers of Dehn twists that normally generate $\mathrm{Mod}(S_g)$

Genetic-fuzzy based load balanced protocol for WSNs

Recent advancement in wireless sensor networks primarily depends upon energy constraint. Clustering is the most effective energy-efficient technique to provide robust, fault-tolerant and also enhance network lifetime and coverage. Selection of optimal number of cluster heads and balancing the load of cluster heads are most challenging issues. Evolutionary based approach and soft computing approach are best suitable for counter the above problems rather than mathematical approach. In this paper we propose hybrid technique where Genetic algorithm is used for the selection of optimal number of cluster heads and their fitness value of chromosome to give optimal number of cluster head and minimizing the energy consumption is provided with the help of fuzzy logic approach. Finally cluster heads uses multi-hop routing based on A*(A-star) algorithm to send aggregated data to base station which additionally balance the load. Comparative study among LEACH, CHEF, LEACH-ERE, GAEEP shows that our proposed algorithm outperform in the area of total energy consumption with various rounds and network lifetime, number of node alive versus rounds and packet delivery or packet drop ratio over the rounds, also able to balances the load at cluster head

Generating the liftable mapping class groups of cyclic covers of spheres

For $g\geq 2$, let $\mathrm{Mod}(S_g)$ be the mapping class group of closed orientable surface $S_g$ of genus $g$. In this paper, we derive a finite generating set for the liftable mapping class groups corresponding to finite-sheeted regular branched cyclic covers of spheres. As an application, we provide an algorithm to derive presentations of these liftable mapping class groups, and the normalizers and centralizers of periodic mapping classes corresponding to these covers. Furthermore, we determine the isomorphism classes of the normalizers of irreducible periodic mapping classes in $\mathrm{Mod}(S_g)$. Moreover, we derive presentations for the liftable mapping class groups corresponding to covers induced by certain reducible periodic mapping classes. Consequently, we derive a presentation for the centralizer and normalizer of a reducible periodic mapping class in $\mathrm{Mod}(S_g)$ of the highest order $2g+2$. As final applications of our results, we recover the generating sets of the liftable mapping class groups of the hyperelliptic cover obtained by Birman-Hilden and the balanced superelliptic cover obtained by Ghaswala-Winarski.Comment: 1 figur

Infinite metacyclic subgroups of the mapping class group

For $g\geq 2$, let $\text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g$. In this paper, we provide necessary and sufficient conditions for the existence of infinite metacyclic subgroups of $\text{Mod}(S_g)$. In particular, we provide necessary and sufficient conditions under which a pseudo-Anosov mapping class generates an infinite metacyclic subgroup of $\text{Mod}(S_g)$ with a nontrivial periodic mapping class. As applications of our main results, we establish the existence of infinite metacyclic subgroups of $\text{Mod}(S_g)$ isomorphic to $\mathbb{Z}\rtimes \mathbb{Z}_m, \mathbb{Z}_n \rtimes \mathbb{Z}$, and $\mathbb{Z} \rtimes \mathbb{Z}$. Furthermore, we derive bounds on the order of a nontrivial periodic generator of an infinite metacyclic subgroup of $\text{Mod}(S_g)$ that are realized. Finally, we show that the centralizer of an irreducible periodic mapping class $F$ is either $\langle F\rangle$ or $\langle F\rangle \times \langle i\rangle$, where $i$ is a hyperelliptic involution.Comment: 25 pages, 18 figure

Goal setting for improvement in product development performance of organizations

Thesis (S.M. in Engineering and Management)--Massachusetts Institute of Technology, Engineering Systems Division, February 2013.Cataloged from PDF version of thesis. "Sep 2012."Includes bibliographical references (pages 107-109).Companies have been constantly trying for ways and means to improve R&D performance as it is one of the most important competitive advantage tools of an organization. Literature review on R&D performance improvement suggests that, lot of focus is on measuring R&D performance and on specific problem solving approaches like six sigma and lean. Frameworks like capability maturity model integration (CMMI) and product development self-assessment tool (PDSAT) provide holistic performance assessment, but fall short on providing clear guidance for performance improvement interventions. Goal setting theory, a proven theory that is widely applied in individual performance improvement has got limited attention in R&D performance improvement approaches and frameworks. Practitioners in the industry point to the need for goal setting in R&D and identify that as a gap in current performance improvement methodologies. This thesis attempts to fill this gap by proposing DEAL framework, a practical approach for defining future goals in R&D performance improvement efforts.by Pankaj Kumar Kashyap.S.M.in Engineering and Managemen

Cosmological Implications of Unimodular Gravity

We consider a model of gravity and matter fields which is invariant only under unimodular general coordinate transformations (GCT). The determinant of the metric is treated as a separate field which transforms as a scalar under unimodular GCT. Furthermore we also demand that the theory is invariant under a new global symmetry which we call generalized conformal invariance. We study the cosmological implications of the resulting theory. We show that this theory gives a fit to the high-z supernova data which is identical to the standard Big Bang model. Hence we require some other cosmological observations to test the validity of this model. We also consider some models which do not obey the generalized conformal invariance. In these models we can fit the supernova data without introducing the standard cosmological constant term. Furthermore these models introduce only one dark component and hence solve the coincidence problem of dark matter and dark energy.Comment: 18 pages, no figures, major revisions, substantial changes in analysis, results and conclusion