Asymptotic Counting Formulas for Markoff-Hurwitz Tuples

The Markoff equation is a Diophantine equation in 3 variables first studied in Markoff\u27s celebrated work on indefinite binary quadratic forms. We study the growth of solutions to an n variable generalization of the Markoff equation, which we refer to as the Markoff-Hurwitz equation. We prove explicit asymptotic formulas counting solutions to this generalized equation with and without a congruence restriction. After normalizing and linearizing the equation, we show that all but finitely many solutions appear in the orbit of a certain semigroup of maps acting on finitely many root solutions. We then pass to an accelerated subsemigroup of maps for which the dynamical system is uniformly contracting and show that both asymptotic formulas are consequences of a general asymptotic formula for a vector-valued counting function related to this accelerated dynamical system. This general formula is obtained using methods from symbolic dynamics, following a technique due to Lalley

Generalized Ramanujan Primes

In 1845, Bertrand conjectured that for all integers $x\ge2$, there exists at least one prime in $(x/2, x]$. This was proved by Chebyshev in 1860, and then generalized by Ramanujan in 1919. He showed that for any $n\ge1$, there is a (smallest) prime $R_n$ such that $\pi(x)- \pi(x/2) \ge n$ for all $x \ge R_n$. In 2009 Sondow called $R_n$ the $n$th Ramanujan prime and proved the asymptotic behavior $R_n \sim p_{2n}$ (where $p_m$ is the $m$th prime). In the present paper, we generalize the interval of interest by introducing a parameter $c \in (0,1)$ and defining the $n$th $c$-Ramanujan prime as the smallest integer $R_{c,n}$ such that for all $x\ge R_{c,n}$, there are at least $n$ primes in $(cx,x]$. Using consequences of strengthened versions of the Prime Number Theorem, we prove that $R_{c,n}$ exists for all $n$ and all $c$, that $R_{c,n} \sim p_{\frac{n}{1-c}}$ as $n\to\infty$, and that the fraction of primes which are $c$-Ramanujan converges to $1-c$. We then study finer questions related to their distribution among the primes, and see that the $c$-Ramanujan primes display striking behavior, deviating significantly from a probabilistic model based on biased coin flipping; this was first observed by Sondow, Nicholson, and Noe in the case $c = 1/2$. This model is related to the Cramer model, which correctly predicts many properties of primes on large scales, but has been shown to fail in some instances on smaller scales.Comment: 13 pages, 2 tables, to appear in the CANT 2011 Conference Proceedings. This is version 2.0. Changes: fixed typos, added references to OEIS sequences, and cited Shevelev's preprin

ProteinParser:a Community Based Tool for the Generation of a Detailed Protein Consensus and FASTA Output

Comparison of bioinformatic data is a common application in the life sciences and beyond. In this communication, a novel Java based software tool, ProteinParser, is outlined. This soft- ware tool calculates a detailed consensus, or most common, amino acid at a given position in an aligned protein set, whilst also generating a full consensus protein FASTA output. A second application of this software tool, computing a consensus amino acid given a toler- ance threshold, is also demonstrated. The phytase and the common bacterial �-lactamase proteins are analysed as ‘proof of concept’ examples. Consensus proteins, as generated by ProteinParser, are regularly utilised in the selection of residues for protein stabilisation muta- genesis; however, this widely applicable software tool will find many alternative applications in areas such as protein homology modelling

An asymptotic formula for integer points on Markoff-Hurwitz varieties

We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation x21+x22+⋯+x2n=ax1x2⋯xn+k. When n≥4, the previous best result is by Baragar (1998) that gives an exponential rate of growth with exponent β that is not in general an integer when n≥4. We give a new interpretation of this exponent of growth in terms of the unique parameter for which there exists a certain conformal measure on projective space

The epidemiology of UK autoimmune liver disease varies with geographic latitude

BACKGROUND & AIMS: The epidemiology of autoimmune liver disease (AILD) is challenging to study because of the diseases’ rarity and because of cohort selection bias. Increased incidence farther from the Equator has been reported for multiple sclerosis, another autoimmune disease. We assessed the incidence of primary biliary cholangitis (PBC), primary sclerosing cholangitis (PSC), and autoimmune hepatitis (AIH) in relation to latitude. METHODS: We performed a retrospective cohort study using anonymized UK primary care records from January 1, 2002, to May 10, 2016. All adults without a baseline diagnosis of AILD were included and followed up until the first occurrence of an AILD diagnosis, death, or they left the database. Latitude was measured as registered general practice rounded down to whole degrees. RESULTS: The cohort included 8,590,421 records with 53.3 × 10(7) years of follow-up evaluation from 694 practices. There were 1314 incident cases of PBC, 396 of PSC, and 1034 of AIH. Crude incidences were as follows: PBC, 2.47 (95% CI, 2.34–2.60); PSC, 0.74 (95% CI, 0.67–0.82); and AIH, 1.94 (95% CI, 1.83–2.06) per 100,000 per year. PBC incidence correlated with female sex, smoking, and deprivation; PSC incidence correlated with male sex and non-smoking; AIH incidence correlated with female sex and deprivation. A more northerly latitude was associated strongly with incidence of PBC: 2.16 (95% CI, 1.79–2.60) to 4.86 (95% CI, 3.93–6.00) from 50°N to 57°N (P = .002) and incidence of AIH: 2.00 (95% CI, 1.65–2.43) to 3.28 (95% CI, 2.53–4.24) (P = .003), but not incidence of PSC: 0.82 (95% CI, 0.60–1.11) to 1.02 (95% CI, 0.64–1.61) (P = .473). Incidence after adjustment for age, sex, smoking, and deprivation status showed similar positive correlations for PBC and AIH with latitude, but not PSC. Incident AIH cases were younger at more northerly latitude. CONCLUSIONS: We describe an association in the United Kingdom between more northerly latitude and the incidence of PBC and AIH that requires both confirmation and explanation

Can routinely collected primary care data be used to predict future risk of morbidity and mortality in newly-diagnosed type 2 diabetes mellitus?

Background/clinical context: Type 2 diabetes (T2DM) is associated with an increased risk of adverse outcomes. Data routinely recorded in general practice electronic patient records could be used to develop risk prediction models to identify those at higher risk and target preventative treatment. Objective: To develop models to predict the 5-year risk of coronary heart disease (CHD), stroke, chronic kidney disease (CKD), and all-cause mortality following a diagnosis of T2DM. Methods: Newly diagnosed T2DM patients registered at a practice contributing data to a large UK general practice database were included in the analyses. The models included clinical predictors routinely recorded following diabetes diagnosis plus cardiovascular preventative treatments. Results: 20041 patients diagnosed with T2DM were included. The proportion of variation explained by each model (R2) was: CHD 0.09; stroke 0.35; CKD 0.34; and mortality 0.58. Hazard ratios for modifiable risks in the mortality model were: current smoking 1.65; blood pressure (high/treated) 1.07; and glycaemic control (HbA1C/%) 1.09 (p<0.01 apart from BP). Conclusion: The models were predictive, particularly for mortality, and suggest that older, male, smokers, those with poor blood pressure and glycaemic control and those with cardiovascular co-morbidity are at highest risk and should be targeted at the point of diagnosis