In the garden of numbers, I will play

This work is an exploration of certain questions pertaining to the sequence 1, 2, 3, 4, 5, .... It is divided into three parts. The first part stems from a project conducted in 2007, headed by K. Ramachandra (Professor at the National Institute of Advanced Studies). We will give the details of the project here. Godfrey Harold Hardy remarked, in his lectures on Srinivasa Ramanujan’s life that Ramanujan, when he was a child in school, discovered that eiθ = cos(θ) + i sin(θ) (from which the relation eiπ +1 = 0 comes as a consequence). He did this entirely on his own. The exact age at which he made this discovery is not known, but Hardy places his age between 7 and 16. In the book, The Music of the Primes, Marcus du Sautoy says that, after this discovery, Ramanujan “found out a few days later that Euler had beaten him to this great discovery by some hundred and fifty years. Humbled and dispirited, Ramanujan hid his calculations in the roof of his house.” Ramanujan’s original proof and method are unknown. Ramanujan had no access to any material of modern mathematics save one, Sidney Luxton Loney’s Trigonometry. It is a mystery how Ramanujan was able to make use of any ideas of complex numbers with the nuances necessary to reach Euler’s equation. What resulted from this project was an introductory pamphlet on Trigonometry, where we begin the simple equations (a+b)2 = a2 + 2ab+b2 or (a − b)2 = a2 − 2ab + b2 (one can start from either), and reach Euler’s formula. This is the probable proof of Ramanujan. This has been published in Mathematics Student. The second part of the thesis is a simplification of many important problems in the analytic theory of numbers. Here, we provide a discussion of many of the current aspects of the theory, such as the Riemann zeta function and the Lindel¨of hypothesis. and explain them in elementary terms. We will also discuss a paper of ours published in the Hardy Ramanujan Journal and relate it to these problems. The third part, which is new, is a generalization of a problem posed by Paul Erd˝os in 1993 in American Mathematical Monthly. Consider the equation n! = a!b! where n>b>a> 1. This equation has an infinite number of solutions. A trivial solution can be constructed as follows: For any arbitrary natural number a, n = a!, and b = a! − 1. Nontrivial solutions exist, for example, 10! = 6!7!. Is there a finite number of non-trivial solutions? In a paper by Florian Luca it was proven conditionally using a weaker form of the famous “ABC conjecture,” that there is a finite number of nontrivial solutions: but the “ABC conjecture” – being a relative of the Riemann hypothesis – may be a long way off from proving. The question reduces to finding a bound on the distance between n and b. A trivial solution would be of the form n − b = 1. Erd˝os was able to obtain an upper bound of the difference to 5 log log n. We improve the absolute constant to 1+� log 2 and generalize this result, with the same constant, to the equation n! = a1!a2! ...ak!, for finite k. This result has been accepted for publication in the Russian journal Matematicheskie Zametki. We proceed with a modification of this theorem where we obtain comparable bound when we substitute the factorial function with the product of terms in a class of arithmetic progressions, that is d(2d)(3d)(md) in the place of m!: The bound we obtain is 1+q/log p where p is the greatest prime factor of the common difference in the arithmetic progression

Huffman Coding as a Non-linear Dynamical System

In this paper, source coding or data compression is viewed as a measurement problem. Given a measurement device with fewer states than the observable of a stochastic source, how can one capture the essential information? We propose modeling stochastic sources as piecewise linear discrete chaotic dynamical systems known as Generalized Lur\"{o}th Series (GLS) which dates back to Georg Cantor's work in 1869. The Lyapunov exponent of GLS is equal to the Shannon's entropy of the source (up to a constant of proportionality). By successively approximating the source with GLS having fewer states (with the closest Lyapunov exponent), we derive a binary coding algorithm which exhibits minimum redundancy (the least average codeword length with integer codeword lengths). This turns out to be a re-discovery of Huffman coding, the popular lossless compression algorithm used in the JPEG international standard for still image compression.Comment: 7 pages, 5 figure

A Remark on a theory of A.E. Ingham

Referring to a theorem of A. E. Ingham, that for all N≥N0 (an absolute constant), the inequality N3≤p≤(N+1)3 is solvable in a prime p, we point out in this paper that it is implicit that he has actually proved that π(x+h)−π(x)∼h(logx)−1 where h=xc and c(>58) is any constant. Further, we point out that even this stronger form can be proved without using the functional equation of ζ(s)

National Consultation on Water Conflicts in India: The State, the People and the Future (NIAS Report No. R2-2010)

The last three decades have seen the gradual erosion of the state's perceived role as a promoter of development and as a steward of natural resources, including water. Both communitarians and neo-liberals coming from different theoretical and methodological angles have converged in their critique of the state. While the communitarians argue for supporting, what are perceived as, old traditions of sustainable community-based resource management, the neo-liberals argue for market-based mechanisms for efficient resource allocation and use. Despite these critiques the state remains an important player in the field of resource management, especially that of water, in developing countries such as India

Estimation of protein requirements in Indian pregnant women using a whole-body potassium counter

Background: The 2007 World Health Organization/Food and Agriculture Organization/United Nations University (WHO/FAO/UNU) recommendation for the Estimated Average Requirement (EAR) of additional protein during pregnancy for a gestational weight gain (GWG) of 12 kg (recalculated from a GWG of 13.8 kg) is 6.7 and 21.7 g/d in the second and the third trimester, respectively. This EAR is based on measurements of potassium accretion in high-income country (HIC) pregnant women. It is not known if low- to middle-income country, but well-nourished, pregnant women have comparable requirements. Objective: We aimed to estimate total body potassium (TBK) accretion during pregnancy in Indian pregnant women, using a whole-body potassium counter (WBKC), to measure their additional protein EAR. Methods: Well-nourished pregnant women (20–40 y, n = 38, middle socioeconomic stratum) were recruited in the first trimester of pregnancy. Anthropometric, dietary, and physical activity measurements, and measurements of TBK using a WBKC, were performed at each trimester and at birth. Results: The mid-trimester weight gain was 2.7 kg and 8.0 kg in the second and the third trimester, respectively, for an average 37-wk GWG of 10.7 kg and a mean birth weight of 3.0 kg. Protein accretion was 2.7 and 5.7 g/d, for an EAR of 8.2 and 18.9 g/d in the second and the third trimester, respectively. The additional protein EAR, calculated for a GWG of 12 kg, was 9.1 and 21.2 g/d in the second and the third trimester, respectively. Conclusion: The additional protein requirements of well-nourished Indian pregnant women for a GWG of 12 kg in the second and third trimesters were similar to the recalculated 2007 WHO/FAO/UNU requirements for 12 kg

Global injury morbidity and mortality from 1990 to 2017 : results from the Global Burden of Disease Study 2017

Correction:Background Past research in population health trends has shown that injuries form a substantial burden of population health loss. Regular updates to injury burden assessments are critical. We report Global Burden of Disease (GBD) 2017 Study estimates on morbidity and mortality for all injuries. Methods We reviewed results for injuries from the GBD 2017 study. GBD 2017 measured injury-specific mortality and years of life lost (YLLs) using the Cause of Death Ensemble model. To measure non-fatal injuries, GBD 2017 modelled injury-specific incidence and converted this to prevalence and years lived with disability (YLDs). YLLs and YLDs were summed to calculate disability-adjusted life years (DALYs). Findings In 1990, there were 4 260 493 (4 085 700 to 4 396 138) injury deaths, which increased to 4 484 722 (4 332 010 to 4 585 554) deaths in 2017, while age-standardised mortality decreased from 1079 (1073 to 1086) to 738 (730 to 745) per 100 000. In 1990, there were 354 064 302 (95% uncertainty interval: 338 174 876 to 371 610 802) new cases of injury globally, which increased to 520 710 288 (493 430 247 to 547 988 635) new cases in 2017. During this time, age-standardised incidence decreased non-significantly from 6824 (6534 to 7147) to 6763 (6412 to 7118) per 100 000. Between 1990 and 2017, age-standardised DALYs decreased from 4947 (4655 to 5233) per 100 000 to 3267 (3058 to 3505). Interpretation Injuries are an important cause of health loss globally, though mortality has declined between 1990 and 2017. Future research in injury burden should focus on prevention in high-burden populations, improving data collection and ensuring access to medical care.Peer reviewe

Measuring universal health coverage based on an index of effective coverage of health services in 204 countries and territories, 1990–2019 : A systematic analysis for the Global Burden of Disease Study 2019

Background Achieving universal health coverage (UHC) involves all people receiving the health services they need, of high quality, without experiencing financial hardship. Making progress towards UHC is a policy priority for both countries and global institutions, as highlighted by the agenda of the UN Sustainable Development Goals (SDGs) and WHO's Thirteenth General Programme of Work (GPW13). Measuring effective coverage at the health-system level is important for understanding whether health services are aligned with countries' health profiles and are of sufficient quality to produce health gains for populations of all ages. Methods Based on the Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) 2019, we assessed UHC effective coverage for 204 countries and territories from 1990 to 2019. Drawing from a measurement framework developed through WHO's GPW13 consultation, we mapped 23 effective coverage indicators to a matrix representing health service types (eg, promotion, prevention, and treatment) and five population-age groups spanning from reproductive and newborn to older adults (≥65 years). Effective coverage indicators were based on intervention coverage or outcome-based measures such as mortality-to-incidence ratios to approximate access to quality care; outcome-based measures were transformed to values on a scale of 0–100 based on the 2·5th and 97·5th percentile of location-year values. We constructed the UHC effective coverage index by weighting each effective coverage indicator relative to its associated potential health gains, as measured by disability-adjusted life-years for each location-year and population-age group. For three tests of validity (content, known-groups, and convergent), UHC effective coverage index performance was generally better than that of other UHC service coverage indices from WHO (ie, the current metric for SDG indicator 3.8.1 on UHC service coverage), the World Bank, and GBD 2017. We quantified frontiers of UHC effective coverage performance on the basis of pooled health spending per capita, representing UHC effective coverage index levels achieved in 2019 relative to country-level government health spending, prepaid private expenditures, and development assistance for health. To assess current trajectories towards the GPW13 UHC billion target—1 billion more people benefiting from UHC by 2023—we estimated additional population equivalents with UHC effective coverage from 2018 to 2023. Findings Globally, performance on the UHC effective coverage index improved from 45·8 (95% uncertainty interval 44·2–47·5) in 1990 to 60·3 (58·7–61·9) in 2019, yet country-level UHC effective coverage in 2019 still spanned from 95 or higher in Japan and Iceland to lower than 25 in Somalia and the Central African Republic. Since 2010, sub-Saharan Africa showed accelerated gains on the UHC effective coverage index (at an average increase of 2·6% [1·9–3·3] per year up to 2019); by contrast, most other GBD super-regions had slowed rates of progress in 2010–2019 relative to 1990–2010. Many countries showed lagging performance on effective coverage indicators for non-communicable diseases relative to those for communicable diseases and maternal and child health, despite non-communicable diseases accounting for a greater proportion of potential health gains in 2019, suggesting that many health systems are not keeping pace with the rising non-communicable disease burden and associated population health needs. In 2019, the UHC effective coverage index was associated with pooled health spending per capita (r=0·79), although countries across the development spectrum had much lower UHC effective coverage than is potentially achievable relative to their health spending. Under maximum efficiency of translating health spending into UHC effective coverage performance, countries would need to reach $1398 pooled health spending per capita (US$ adjusted for purchasing power parity) in order to achieve 80 on the UHC effective coverage index. From 2018 to 2023, an estimated 388·9 million (358·6–421·3) more population equivalents would have UHC effective coverage, falling well short of the GPW13 target of 1 billion more people benefiting from UHC during this time. Current projections point to an estimated 3·1 billion (3·0–3·2) population equivalents still lacking UHC effective coverage in 2023, with nearly a third (968·1 million [903·5–1040·3]) residing in south Asia. Interpretation The present study demonstrates the utility of measuring effective coverage and its role in supporting improved health outcomes for all people—the ultimate goal of UHC and its achievement. Global ambitions to accelerate progress on UHC service coverage are increasingly unlikely unless concerted action on non-communicable diseases occurs and countries can better translate health spending into improved performance. Focusing on effective coverage and accounting for the world's evolving health needs lays the groundwork for better understanding how close—or how far—all populations are in benefiting from UHC