Mathematics

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In this paper, we prove an extension of the Helson–Lowdenslager theorem for L2(T) for sub - Hilbert spaces under certain axioms. Furthermore, we provide an alternate proof of characterization of shift invariant sub-Hilbert spaces contained in H2(D) satisfying similar axioms. © 2021, Tusi Mathematical Research Group (TMRG).

World University root node

(SoNS) School of Natural Sciences

Advances in Operator Theory

On extension of Beurling–Helson–Lowdenslager theorem

Non-pharmaceutical interventions (NPIs) involving social-isolation strategies such as self-quarantine (SQ) and social-distancing (SD) are useful in controlling the spread of infections that are transmitted through human-to–human contacts, e.g., respiratory diseases such as COVID-19. In the absence of a safe and effective cure or vaccine during the first ten months of the COVID-19 pandemic, countries around the world implemented these social-isolation strategies and other NPIs to reduce COVID-19 transmission. But, individual and public perception play a crucial role in the success of any social-isolation measure. Thus, in spite of governments’ initiatives to use NPIs to combat COVID-19 in many countries around the world, individual choices rendered social-isolation unsuccessful in some of these countries. This resulted in huge outbreaks that imposed a substantial morbidity, mortality, hospitalization, economic, etc., toll on human lives. In particular, human choices pose serious challenges to public health strategic decision-making in controlling the COVID-19 pandemic. To unravel the impact of this behavioral response to social-isolation on the burden of the COVID-19 pandemic, we develop a model framework that integrates COVID-19 transmission dynamics with a multi-strategy evolutionary game approach of individual decision-making. We use this integrated framework to characterize the evolution of human choices in social-isolation as the disease progresses and public health control measures such as mandatory lockdowns are implemented. Analysis of the model illustrates that SD plays a major role in reducing the burden of the disease compared to SQ. Parameter estimation using COVID-19 incidence data, as well as different lockdown data sets from India, and scenario analysis involving a combination of Voluntary-Mandatory implementation of SQ and SD shows that the effectiveness of this approach depends on the type of isolation, and the time and period of implementation of the selected isolation measure during the outbreak. © 2021 Elsevier Ltd

Journal of Theoretical Biology

Human choice to self-isolate in the face of the COVID-19 pandemic: A game dynamic modelling approach

Climate changes and environmental stresses have a consequential association with crop plant growth and yield, meaning it is necessary to cultivate crops that have tolerance toward the changing climate and environmental disturbances such as water stress, temperature fluctuation, and salt toxicity. Recent studies have shown that trans-acting regulatory elements, including microRNAs (miRNAs) and transcription factors (TFs), are emerging as promising tools for engineering naive improved crop varieties with tolerance for multiple environmental stresses and enhanced quality as well as yield. However, the interwoven complex regulatory function of TFs and miRNAs at transcriptional and post-transcriptional levels is unexplored in Oryza sativa. To this end, we have constructed a multiple abiotic stress responsive TF-miRNA-gene regulatory network for O. sativa using a transcriptome and degradome sequencing data meta-analysis approach. The theoretical network approach has shown the networks to be dense, scale-free, and small-world, which makes the network stable. They are also invariant to scale change where an efficient, quick transmission of biological signals occurs within the network on extrinsic hindrance. The analysis also deciphered the existence of communities (cluster of TF, miRNA, and genes) working together to help plants in acclimatizing to multiple stresses. It highlighted that genes, TFs, and miRNAs shared by multiple stress conditions that work as hubs or bottlenecks for signal propagation, for example, during the interaction between stress-responsive genes (TFs/miRNAs/other genes) and genes involved in floral development pathways under multiple environmental stresses. This study further highlights how the fine-tuning feedback mechanism works for balancing stress tolerance and how timely flowering enable crops to survive in adverse conditions. This study developed the abiotic stress-responsive regulatory network, APRegNet database (http://lms.snu.edu.in/APRegNet), which may help researchers studying the roles of miRNAs and TFs. Furthermore, it advances current understanding of multiple abiotic stress tolerance mechanisms. © Copyright © 2021 Sharma, Upadhyay, Bhattacharya and Singh.

Life Sciences

Fulltext

Frontiers in Genetics

Abiotic Stress-Responsive miRNA and Transcription Factor-Mediated Gene Regulatory Network in Oryza sativa: Construction and Structural Measure Study

For a unital C*-algebra A and a subspace B of A, a characterization for a best approximation to an element of A in B is obtained. As an application, a formula for the distance of an element of A from B has been obtained, when a best approximation of that element to B exists. Further, a characterization for Birkhoff-James orthogonality of an element of a Hilbert C*-module to a subspace is obtained. © 2021 Ramanujan Mathematical Society. All rights reserved.

Journal of the Ramanujan Mathematical Society

Best approximations, distance formulas and orthogonality in C**-algebras

Flu is an acute respiratory infection caused by the influenza virus. The outbreak usually occurs every year in temperate region during the fall and winter seasons, but it is present year-round in tropics. Perceived risk of infection, affordability and lack of awareness among the population results in a low level of vaccination coverage. To control disease transmission and promote vaccination, public health officials use media coverage to spread awareness on vaccine safety, vaccine coverage, disease prevalence in the population through public health websites, advertisements, and other social media web pages. Media coverage acts as an incentive as it helps to decrease overall transmission potential and also at the same time increases the vaccination coverage in the population. Since the public health department has a limited budget, it needs to make optimum allocation of its effort to reduce the total cost of infection. Our paper investigates the effect of media coverage using SIR model of disease transmission. We look at three possible functional relationships - linear, exponential, and hyperbolic - the way media coverage may affect the disease transmission and vaccination rate. We derive necessary conditions of optimal solution using Optimal Control Theory and Pontryagin Maximum Principle (PMP) to minimize the total cost for infection. Analysis of our paper demonstrates that the cost of optimal management is four times less than the cost of constant control effort, and putting more effort into reducing transmission is optimal rather than an effort to increase vaccination at the beginning of the outbreak. Analysis of the role of media coverage under three different scenarios may help in formulating policies for public health programs in mitigating the influenza outbreak. © 2021 World Scientific Publishing Company.

Journal of Biological Systems

ROLE of MEDIA COVERAGE in MITIGATING AN EPIDEMIC OUTBREAK: AN OPTIMAL CONTROL MODEL

Novel strains emerge every year and cause seasonal flu epidemic. These variants exhibit differences in antigenicity and may escape immunity from vaccines and thus invade the population. Here, we devise a discrete population game model of vaccination and transmission of two strains–a resident and a relatively rare mutant–to explain the influence of social behaviour, specifically vaccinating behaviour on the emergence of the mutant strain in the population. Our integrated framework of behaviour-prevalence model exhibits time evolution of two strategic interactions such as individuals' perceived vaccination risk and probability of infection, where both declines as more and more individuals vaccinate in the population. Numerical simulation and analysis of the model show that the predominant Nash strategy allows invasion of the mutant strain under very disparate individuals perceived disease risks from these two pathogens and relatively lower vaccine efficacy against the mutant. The model also demonstrates that invasion of the mutant has a nonlinear dependence on epidemiological characteristics such as the relative force of infection, cross-immunity, perceived cost of treatment and vaccine efficacy. This study exemplifies that adaptive social behaviour is an important component of public health strategic decision-making during the control of the flu-like epidemic. © 2021 Informa UK Limited, trading as Taylor &amp; Francis Group.

Journal of Difference Equations and Applications

A discrete population game model of vaccination exploring the role of individuals' risk perceptions in the invasion of a mutant strain

In the last few decades, the concept of Birkhoff–James orthogonality has been used in several applications. In this survey article, the results known on the necessary and sufficient conditions for Birkhoff–James orthogonality in certain Banach spaces are mentioned. Their applications in studying the geometry of normed spaces are given. The connections between this concept of orthogonality, and the Gateaux derivative and the subdifferential set of the norm function are provided. Several interesting distance formulas can be obtained using the characterizations of Birkhoff–James orthogonality, which are also mentioned. In the end, some new results are obtained. © 2021, Springer Nature Switzerland AG.

Operator Theory: Advances and Applications

Birkhoff–James Orthogonality and Applications: A Survey

Mathematics has been applied to problems involving loans and investments from very early times. The Mesopotamians were calculating compound interest around 2000 BC. Later, in India and China, problems of debt were used to motivate negative numbers. The connections have only strengthened with time and the flow of ideas has been especially rich in the last century. We give a brief history of these connections and also explore the role of finance in the teaching of mathematics and as a career for mathematicians.

Proceedings of the Indian National Science Academy

The Interplay of Mathematics and Finance

In this article, we prove various properties of the Calkin-Wilf tree. We also see how the Minkowski question mark function acts on the Calkin-Wilf tree and its diagonals. © 2020, Indian Academy of Sciences.

Resonance

Calkin-Wilf Tree

Let 0 → a → e → g → 0 be an abelian extension of Lie superalgebras. In this article, corresponding to this extension we construct two exact sequences connecting the various automorphism groups and the 0-th homogeneous part of the second cohomology group, H2(g, a)0 . These exact sequences constitute an analogue of the well-known Wells exact sequence for group extensions. From this it follows that the obstruction for a pair of automorphism (ℙ, ℙ) 2 Aut(a) Aut(g) to be induced from an automorphism in Auta(e) lies in H2(g, a)0 . Then we consider the family of Heisenberg Lie superalgebras and show that not all pairs are inducible in this family. We also give some necessary and sufficient conditions for inducibility of pairs arising in this family. ©2020 Heldermann Verlag.