Mathematics

Shiv Nadar University is a comprehensive, multidisciplinary, research-focused, and student-centric University offering a full range of academic programs at the undergraduate, postgraduate and doctoral level. With state-of-the-art infrastructure, the University comprises of academic wings, sophisticated labs, international standard sports facilities, amphitheaters, auditorium, conference rooms and smart classrooms. The University&rsquo;s goal is to become internationally recognized for the quality of its research and creative endeavors and their applicability to improving quality of life, generating new insights and expanding the boundaries of human knowledge creativity. Committed to excellence in teaching, research and service, the University aims to serve the higher education needs of India and the world beyond.

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Recently Yousefi and Hesameddini [13] have obtained a characterization for shift invariant subspaces of a special class of Hilbert spaces contained in the Hardy space H2. In the present note we settle an open problem posed by them in their paper. In fact, by discussing invariance under multiplication by finite Blaschke factors we prove a far more general result than the main result of [13]. We prove our results under much weaker assumptions than the assumptions of [13] and with a simpler proof. © 2011 The Japan Academy.

(SoNS) School of Natural Sciences

World University root node

Proceedings of the Japan Academy Series A: Mathematical Sciences

Invariant subspaces of certain sub hilbert spaces of H2

This book is broadly about the mathematical aspects of finance. It introduces the reader to the basic concepts and products of modern finance and explores various mathematical models dealing with quantification of risk, which form the backbone of modern financial analysis. The emphasis is not so much on the details of the financial world as the basic principles by which one seeks an understanding of it. No prior knowledge of economics or finance is called for—an exposure to basic calculus and probability is all that is required of the reader. The appendix covers this ground in fair detail and would itself serve as a comprehensive primer of mathematics for finance for a beginner. The book is peppered with examples that use real-life data to ground the theory covered in the book. The exercises to be worked out are also interspersed in the text—their purpose varies from simple practice in applying formulas to extending the ideas learnt to new situations. Solutions to all the exercise problems are included as Appendix C, a feature that will be welcomed by both students and faculty. The book will serve well as an introductory book on applied mathematics in finance, of interest to students of mathematics, finance and financial management. For those starting out as practitioners of mathematical finance, this is an ideal introduction.

The Calculus of Finance

Several studies have found that some parents delay the age at which their children receive pediatric vaccines due to perception of higher vaccine risk at the recommended age of vaccination. This has been particularly apparently during the Measles-Mumps-Rubella scare in the United Kingdom. Under a voluntary vaccination policy, vaccine coverage in certain age groups is a potentially complex interplay between vaccinating behaviour, disease dynamics, and age-specific risk factors. Here, we construct an age-structured game dynamic model, where individuals decide whether to vaccinate according to imitation dynamics depending on age-dependent disease prevalence and perceived risk of vaccination. Individuals may be timely vaccinators, delayers, or non-vaccinators. The model exhibits multiple equilibria and a broad range of possible dynamics. For certain parameter regimes, the proportion of timely vaccinators and delayers oscillate in an anti-phase fashion in response to oscillations in infection prevalence. Under an exogenous change to the perceived risk of vaccination as might occur during a vaccine scare, the model can also capture an increase in delayer strategists similar in magnitude to that observed during the Measles-Mumps-Rubella vaccine scare in the United Kingdom. Our model also shows that number of delayers steadily increases with increasing severity of the scare, whereas it saturates to specific value with increases in duration of the scare. Finally, by comparing the model dynamics with and without the option of a delayer strategy, we show that adding a third delayer strategy can have a stabilizing effect on model dynamics. In an era where individual choice-rather than accessibility-is becoming an increasingly important determinant of vaccine uptake, more infectious disease models may need to use game theory or related techniques to determine vaccine uptake. © 2010 Elsevier Ltd.

Journal of Theoretical Biology

A game dynamic model for delayer strategies in vaccinating behaviour for pediatric infectious diseases

We prove two theorems. The first theorem reduces to a scalar situation the well known vector-valued generalization of the Helson-Lowdenslager theorem that characterizes the invariant subspaces of the operator of multiplication by the coordinate function z on the vector-valued Lebesgue space L2(T ℂn). Our approach allows us to prove an equivalent version of the vector-valued Helson-Lowdenslager theorem in a completely scalar setting, thereby eliminating the use of range functions and partial isometries. The other three major advantages provided by our characterization are: (i) we provide precise necessary and sufficient conditions for the presence of reducing subspaces inside simply invariant subspaces; (ii) we give a complete description of the wandering vectors; (iii) we prove the theorem in the setting of all the Lebesgue spaces Lp (0 &lt; p ≤ ∞). Our second theorem generalizes the first theorem along the lines of de Branges' generalization of Beurling's theorem by characterizing those Hilbert spaces that are simply invariant under multiplication by zn and which are contractively contained in Lp (1 ≤ p ≤ ∞). This also generalizes a theorem of Paulsen and Singh [Proc. Amer. Math. Soc. 129 (2000)] as well as the main theorem of Redett [Bull. London Math. Soc. 37 (2005)]. © Instytut Matematyczny PAN, 2010.

Studia Mathematica

A finite multiplicity Helson-Lowdenslager-de Branges theorem

We study the dynamics of a disease under administration of a vaccine and antiviral drug, where the disease transmits directly from the parents to the offspring (vertical transmission) and also through contact with infective individuals (horizontal transmission). While vaccination to those susceptible reduces the horizontal transmission, administration of the antiviral drug to infected individuals lessens the chance of vertical transmission. Thus the vaccine and antiviral drug play different roles in controlling the disease, which has both vertical and horizontal transmission. We develop a 3D model with Susceptible-Infected-Recovered under vaccination to the susceptible and antiviral treatment to the infected and consider a control theoretic approach using the Pontryagin maximum principle to analyse the cost-effectiveness of the control process. Our results demonstrate that a mixed intervention strategy of vaccination and antiviral drug in a proper ratio is the most effective way to control the disease. We show that cost-effectiveness of both intervention strategies intimately depends on disease-related parameters, such as force of infection, probability of being infected to offspring from infected mothers, loss of immunity or reinfection and also on cost of treatment. © 2010 Taylor & Francis.

Computational and Mathematical Methods in Medicine

Optimal control of vertically transmitted disease: An integrated approach

Background: Mathematical models have formalized how free-rider effects can threaten the stability of high vaccine coverage levels under established voluntary vaccination programs. However, little research has addressed the question of when free-riding begins to develop when a new vaccine is first introduced in a population. Methodology/Principal Findings: Here, we combine a game theoretical model of vaccinating behavior with an agestructured compartmental model to analyze rational vaccinating behavior in the first years of a universal immunization program, where a new vaccine is free to all children of a specified age. The model captures how successive birth cohorts face different epidemiological landscapes that have been shaped by the vaccinating decisions of previous birth cohorts, resulting in a strategic interaction between individuals in different birth cohorts. The model predicts a Nash equilibrium coverage level of 100% for the first few birth cohorts under the new program. However, free-riding behavior emerges very quickly, with the Nash equilibrium vaccine coverage dropping significantly within 2-5 years after program initiation. Subsequently, a rich set of coupled dynamics between infection prevalence and vaccinating behaviors is possible, ranging from relatively stable (but reduced) coverage in later birth cohorts to wide fluctuations in vaccine coverage from one birth cohort to the next. Individual tolerance for vaccine risk also starts out at relatively high levels before dropping significantly within a few years. Conclusions/Significance: These results suggest that even relatively new immunization programs can be vulnerable to drops in vaccine coverage caused by vaccine scares and exacerbated by herd immunity effects, necessitating vigilance from the start.

Fulltext

PLoS ONE

Rapid emergence of free-riding behavior in new pediatric immunization programs

We give a short direct proof of Agler's factorization theorem that uses the Blecher-Ruan-Sinclair characterization of operator algebras. The key ingredient of this proof is an operator algebra factorization theorem. Our proof provides some additional information about these factorizations in the case of polynomials. © 2009 American Mathematical Society.

Proceedings of the American Mathematical Society

An operator algebraic proof of Agler's factorization theorem

The fundamental building blocks of any ecosystem, the food webs, which are assemblages of species through various interconnections, provide a central concept in ecology. The study of a food web allows abstractions of the complexity and interconnectedness of natural communities that transcend the specific details of the underlying systems. For example, Fig. 1 shows a typical food web, where the species are connected through their feeding relationships. The top predator, Heliaster (starfish) feeds on many gastropods like Hexaplex, Morula, Cantharus, etc., some of whom predate on each other [129]. Interactions between species in a food web can be of many types, such as predation, competition, mutualism, commensalism, and ammensalism (see Section 1.1, Fig. 2). © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2009.

Modeling and Simulation in Science, Engineering and Technology

Ecological networks: Structure, interaction strength, and stability

This study deals with control of pest population in agricultural ecosystem using sterile insect technique (SIT). A three-dimensional stage-structured model of the pest under the release of sterile male has been considered. This article also considers the effect of this technique under immigration of wild insects in the control area. Moreover, the deterministic model is extended to a stochastic one allowing random fluctuations around the positive interior equilibrium. The stochastic stability properties of the model are investigated, both analytically and numerically. The thresholds of sterile males that obtained from our study might be helpful to understand and implement the technique properly. Copyright © Taylor & Francis Group, LLC.

Stochastic Analysis and Applications

A stage-structured stochastic model of sterile male under immigration

In this paper, we propose a mathematical model of viral infection in pest control. As the viral infection induces host lysis which releases more virus into the environment, on the average 'κ' viruses per host, κ (1,∞), so the 'virus replication parameter' is chosen as the main parameter on which the dynamics of the infection depends. There exists a threshold value κ0 beyond which the infection persists in the system. Still for increasing the value of κ, the endemic equilibrium bifurcates towards a periodic solution, which essentially indicates that the viral pesticide has a density-dependent 'numerical response' component to its action. Investigation also includes the dependence of the process on predation of natural enemy into the system. A concluding discussion with numerical simulation of the model is also presented. © 2007 Society for Mathematical Biology.