Mathematics

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A bridge to the world of mathematics for readers who want to gain a good foundation in basic mathematical skills for research and other activities.This book aims to help students of social sciences, liberal arts, and humanities to develop the ability to analyze and reason mathematically, to model situations and problems, and to be able to infer, present, and communicate their analysis effectively. Mathematical concepts are presented in both historical and everyday contexts to ease their utilization in the real world.Readers are introduced to the skills of expressing mathematical ideas using the language of sets, logically analyzing arguments and their validity, processing and interpreting data, and using probability to handle the inherent randomness of our world. Chapters dedicated to symmetry, perspective, and art will enable readers to reason, model, and evaluate everyday situations. The book will also increase awareness of how mathematical patterns pervade the world around us.

(SoNS) School of Natural Sciences

World University root node

A Bridge to Mathematics

We report a discrete variant of Farkas’ Lemma in the setting of a module over a linearly ordered commutative ring. The ring may contain zero divisors, and need not be associative nor unital, but we need a certain hypothesis about the ring. Finally, we discuss the result and compare it with other related results found in the literature. © 2017 Elsevier B.V.

Operations Research Letters

A discrete variant of Farkas’ Lemma

We obtain necessary and sufficient conditions for a matrix A to be Birkhoff-James orthogonal to another matrix B in the Ky Fan k-norms. A characterization for A to be Birkhoff-James orthogonal to any subspace W of M(n) is also obtained. © 2016 Informa UK Limited, trading as Taylor & Francis Group.

Fulltext

Linear and Multilinear Algebra

Orthogonality of matrices in the Ky Fan k-norms

Network theoretic approach has been used to model and study the flow of ecological information, growth and connectivity on landscape level of anemochory (wind dispersal) of Himalayan moist temperate forest species in the Western Himalaya region. A network is formally defined and derived for seed dispersion model of target floral species where vertices represent habitat patches which are connected by an edge if the distance between the patches is less than a threshold distance. We define centrality of a network and computationally identify the habitat patches that are central to the process of seed dispersion to occur across the network. These central patches are located on map and geographical regions critically important for the flow of ecological information across the network are identified as Gharwal region and eastern Himachal Pradesh of Indian Himalaya. We find that the network of habitat patches is a scale-free network and at the same time it also displays small-world property characterized by high clustering and low average shortest path length. As a result, ecological information propagates rapidly and evenly on a local scale. Hubs in the network are identified as important centres for dissemination of ecological information (seeds) and need to be conserved against a potential attack by malicious agents and also ecological shocks. The network showcase a well-formed community structure. As a consequence of these structural properties of the network, anemochory floral species studied in this work are likely to thrive across the ecological network of forest patches in the Western Himalaya region over time. © 2017 Elsevier B.V.

Ecological Modelling

A network theoretic study of ecological connectivity in Western Himalayas

SUMMARY Measles elimination goals have been adopted in a range of countries, sub-regions, and regions since the WHO declared an elimination goal by 2015 or 2020. All countries attempt to achieve and maintain high coverage through routine immunization programmes. This routine strategy, however, does not ensure the elimination goal of measles. Many developed countries, such as the United States, that have succeeded in interrupting measles transmission earlier, are now experiencing outbreaks with an increasing number of cases. Using a stochastic, age-structured model of measles vaccination dynamics, we explore and characterize the transient dynamics of measles susceptibility in the years following the implementation of routine vaccination at the herd immunity threshold. We demonstrate how a population could face risks of potentially large outbreaks even within few years of vaccination. We characterize different risk profiles depending on the incidence pattern in the years prior to vaccination. These results suggest that the classic critical vaccination threshold is necessary to achieve herd immunity, but not sufficient to prevent long periods of transient, super-critical dynamics. Our results suggest the need of future work for more careful monitoring of the impacts of current immunization programmes, and developing models that take into account more complicated vaccination strategies, demographic factors, and population movements. Copyright © Cambridge University Press 2016 Â.

Epidemiology and Infection

Age-specific mixing generates transient outbreak risk following critical-level vaccination

This paper describes completely the invariant subspaces of the operator of multiplication by a finite Blaschke factor on the Banach space BMOA of analytic functions with bounded mean oscillation on the unit circle in the complex plane. As a simple application, we describe by very elementary means, the invariant subspaces of the co-analytic Toeplitz operator TB on H1. In the simplest case when B(z) = z, the invariant subspaces of TB on H1 were described by fairly deep arguments until the appearance of an elementary proof by two of the authors (Sahni &amp; Singh). In recent times, the common invariant subspaces of the operators of multiplication by B2 and B3, first in the case of z2 and z3, and then for an arbitrary finite Blaschke B, have proved to be critical in the context of Nevanlinna-Pick type interpolation on H2. Thus, keeping in mind the importance of invariant subspaces, we also offer a characterization of the common invariant subspaces of these operators on BMOA. Our proofs are that much more technical. Again, as an application, we obtain the common invariant subspaces of TB2 and TB3 on the Hardy space H1. © 2017, University at Albany. All rights reserved.

New York Journal of Mathematics

Invariance under finite blaschke factors on BMOA

In this paper, we revisit a result by Jurg et al. (Linear Algebra Appl 141:61–74, 1990) where the necessary and sufficient condition for a bimatrix game to be weakly completely mixed is given. We present an alternate proof of this result using linear complementarity approach. We extend this result to a generalization of bimatrix game introduced by Gowda and Sznajder (Int J Game Theory 25:1–12, 1996) via a generalization of linear complementarity problem introduced by Cottle and Dantzig (J Comb Theory 8:79–90, 1970). We further study completely mixed switching controller stochastic game (in which transition structure is a natural generalization of the single controller games) and extend the results obtained by Filar (Proc Am Math Soc 95:585–594, 1985) for completely mixed single controller stochastic game to completely mixed switching controller stochastic game. A numerical method is proposed to compute a completely mixed strategy for a switching controller stochastic game. © 2016, Springer Science+Business Media New York.

Dynamic Games and Applications

Completely Mixed Strategies for Generalized Bimatrix and Switching Controller Stochastic Game

Perturbation bounds for Mostow's decomposition and the bipolar decomposition of matrices have been computed. To do so, expressions for the derivative of the geometric mean of two positive definite matrices have been derived. © 2017 Elsevier Inc.

Linear Algebra and Its Applications

Perturbation bounds for Mostow's decomposition and the bipolar decomposition

Historically, infectious diseases caused considerable damage to human societies, and they continue to do so today. To help reduce their impact, mathematical models of disease transmission have been studied to help understand disease dynamics and inform prevention strategies. Vaccination–one of the most important preventive measures of modern times–is of great interest both theoretically and empirically. And in contrast to traditional approaches, recent research increasingly explores the pivotal implications of individual behavior and heterogeneous contact patterns in populations. Our report reviews the developmental arc of theoretical epidemiology with emphasis on vaccination, as it led from classical models assuming homogeneously mixing (mean-field) populations and ignoring human behavior, to recent models that account for behavioral feedback and/or population spatial/social structure. Many of the methods used originated in statistical physics, such as lattice and network models, and their associated analytical frameworks. Similarly, the feedback loop between vaccinating behavior and disease propagation forms a coupled nonlinear system with analogs in physics. We also review the new paradigm of digital epidemiology, wherein sources of digital data such as online social media are mined for high-resolution information on epidemiologically relevant individual behavior. Armed with the tools and concepts of statistical physics, and further assisted by new sources of digital data, models that capture nonlinear interactions between behavior and disease dynamics offer a novel way of modeling real-world phenomena, and can help improve health outcomes. We conclude the review by discussing open problems in the field and promising directions for future research. © 2016 Elsevier B.V.

Physics Reports

Statistical physics of vaccination

Topological properties of chemical library networks, such as the average clustering coefficient, average path length, and existence of hubs, can serve as indicators to describe the inherent complexities of chemical libraries. We have used Diversity-Oriented Synthesis (DOS) and Focussed Libraries to investigate the appearance of scale-free properties and absence of small-world behavior in chemical libraries. DOS aims to elicit structural complexity in small compounds with respect to skeleton, functional groups, appendages and stereochemistry. Complexity here indicates incorporation of sp 3 carbons, hydrogen bond acceptors and donors in the molecule. Biological studies have shown how structural complexity enhances the interaction of molecules with complex biological macromolecules. In contrast, Focussed Libraries concentrate on specific scaffolds against a specific biological target. We have quantified the diversity in several DOS and Focussed Libraries based on properties of similarity and dissimilarity threshold networks formed from them. Similarity and dissimilarity networks were generated from diverse chemical libraries at various Tanimoto similarity coefficients (t c) using FP2 and MACCS fingerprints. The dissimilarity networks at very low t c threshold led to the absence of small-world behaviors, as evidenced by low average clustering coefficient and high average path length in comparison to Erdös–Renyi networks. Dissimilarity networks exhibit scale free topology as evidenced by a power law degree distribution. The similarity networks at high t c threshold have shown high clustering coefficients and low average path lengths, without the appearance of hubs. Combining dissimilarity and similarity threshold graphs revealed assortative and dissortative behaviors in the DOS libraries, leading to the conclusion that the vertices of the dissimilarity communities are more likely to share similarity edges, but it is quite unlikely for the vertices in a similarity community to share dissimilarity edges. We propose a simple and convenient diversity quantification tool, QuaLDI (Quantitative Library Diversity Index) to quantify the diversity in DOS and Focussed libraries. We anticipate that these topological properties can be used as descriptors to quantify the diversity in chemical libraries before proceeding for synthesis. © 2016, Springer International Publishing Switzerland.