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Search Results: 1 - 10 of 324 matches for " Byron Heersink "
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Poincaré sections for the horocycle flow in covers of SL(2,R)/SL(2,Z) and applications to Farey fraction statistics
Byron Heersink
Mathematics , 2014,
Abstract: For a given finite index subgroup H of SL(2,Z), we use a process developed by Fisher and Schmidt to lift a Poincar\'e section of the horocycle flow on SL(2,R)/SL(2,Z) found by Athreya and Cheung to the finite cover SL(2,R)/H of SL(2,R)/SL(2,Z). We then use the properties of this section to prove the existence of the limiting gap distribution of various subsets of Farey fractions. Additionally, to each of these subsets of fractions, we extend solutions by Xiong and Zaharescu, and independently Boca, to a Diophantine approximation problem of Erd\H{o}s, Sz\"usz, and Tur\'an.
An effective estimate for the Lebesgue measure of preimages of iterates of the Farey map
Byron Heersink
Mathematics , 2015,
Abstract: Using techniques from infinite ergodic theory, Kessebohmer and Stratmann determined the asymptotic behavior of the Lebesgue measure of sets of the form $F^{-n}[\alpha,\beta]$, where $[\alpha,\beta]\subseteq(0,1]$ and $F$ is the Farey map. In this paper, we provide an effective version of this result, employing mostly basic properties of the transfer operator of the Farey map and an application of Freud's effective version of Karamata's Tauberian theorem.
Continued fraction normality is not preserved along arithmetic progressions
Byron Heersink,Joseph Vandehey
Mathematics , 2015,
Abstract: It is well known that if $0.a_1a_2a_3\dots$ is the base-$b$ expansion of a number normal to base-$b$, then the numbers $0.a_ka_{m+k}a_{2m+k}\dots$ for $m\ge 2$, $k\ge 1$ are all normal to base-$b$ as well. In contrast, given a continued fraction expansion $\langle a_1,a_2,a_3,\dots\rangle$ that is normal (now with respect to the continued fraction expansion), we show that for any integers $m\ge 2$, $k\ge 1$, the continued fraction $\langle a_k, a_{m+k},a_{2m+k},a_{3m+k},\dots\rangle$ will never be normal.
Gap distribution of Farey fractions under some divisibility constraints
Florin P. Boca,Byron Heersink,Paul Spiegelhalter
Mathematics , 2013,
Abstract: For a fixed positive integer d, we show the existence of the limiting gap distribution measure for the sets of Farey fractions a/q of order Q with a not divisible by d, and respectively with q relatively prime with d, as Q tends to infinity.
Cuadernos de Economía , 2010,
Abstract: the objective of the present study is to estimate the economy growth rate in the context of full use of the productive resources. time series univariete models, including a deterministic tendency (with mean breaks), and a space-state model that considers a stochastic tendency and a cyclic component, estimated by kalman′s filter were used. the cyclic component is calculated with and without regime chages (markov-switching models), according to friedman′s idea -plucking model. the results indicate that the chilean economy potential growth rate would be greater than 4% annually, accompanied by economic asymmetric cycles.
Advances Towards Personalised Medicine: What is the Cost of Knowing your Genome?
Kimberley Byron
Opticon1826 , 2011, DOI: 10.5334/opt.111110
Abstract: Since the completion of the Human Genome Project (HGP) in 2003, scientists have been working towards making whole genome sequencing a useful clinical diagnostic tool. The aim of many of the big biotechnology companies is to make this technique affordable and accessible so that it can be routinely used to diagnose rare genetic disorders and tailor medical treatment to an individual’s genetic code – a practice called personalised medicine.
A brief history of muscular dystrophy research: A personal perspective
Kakulas Byron
Neurology India , 2008,
Abstract: The field of myology has undergone remarkable changes. From the period of early clinical descriptions and clinical classifications, new knowledge of these disorders has come from the developments of histopathology, enzyme histochemistry and later, immunocytochemistry and electron microscopy. These techniques have enhanced the understanding of the pathophysiology of myopathies at the cellular level. The parallel evolution of molecular genetics has taken the science further not only by way of understanding and accuracy of diagnosis, but has opened up exciting possibilities of modulation of these chronic debilitating diseases. This review gives a personal perspective of the developments in the field of myology.
Effect of Correlations Between Model Parameters and Nuisance Parameters When Model Parameters are Fit to Data
Byron Roe
Physics , 2013,
Abstract: The effect of correlations between model parameters and nuisance parameters is discussed, in the context of fitting model parameters to data. Modifications to the usual $\chi^2$ method are required. Fake data studies, as used at present, will not be optimum. Problems will occur for applications of the Maltoni-Schwetz \cite{ms} theorem. Neutrino oscillations are used as examples, but the problems discussed here are general ones, which are often not addressed.
Chi-square Fitting When Overall Normalization is a Fit Parameter
Byron Roe
Physics , 2015,
Abstract: The problem of fitting an event distribution when the total expected number of events is not fixed, keeps appearing in experimental studies. In a chi-square fit, if overall normalization is one of the parameters parameters to be fit, the fitted curve may be seriously low with respect to the data points, sometimes below all of them. This problem and the solution for it are well known within the statistics community, but, apparently, not well known among some of the physics community. The purpose of this note is didactic, to explain the cause of the problem and the easy and elegant solution. The solution is to use maximum likelihood instead of chi-square. The essential difference between the two approaches is that maximum likelihood uses the normalization of each term in the chi-square assuming it is a normal distribution, 1/sqrt(2 pi sigma-square). In addition, the normalization is applied to the theoretical expectation not to the data. In the present note we illustrate what goes wrong and how maximum likelihood fixes the problem in a very simple toy example which illustrates the problem clearly and is the appropriate physics model for event histograms. We then note how a simple modification to the chi-square method gives a result identical to the maximum likelihood method.
Fitting theory to data in the presence of background uncertainties
Byron Roe
Physics , 2014,
Abstract: When fitting theory to data in the presence of background uncertainties, the question of whether the spectral shape of the background happens to be similar to that of the theoretical model of physical interest has not generally been considered previously. These correlations in shape are considered in the present note and found to make important corrections to the calculations. The discussion is phrased in terms of $\chi^2$ fits, but the general considerations apply to any fits. Including these new correlations provides a more powerful test for confidence regions. Fake data studies, as used at present, may not be optimum.
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