Transreal limits expose category errors in IEEE 754 floating-point arithmetic and in mathematics

The IEEE 754 standard for oating-point arithmetic is widely used in computing. It is based on real arithmetic and is made total by adding both a positive and a negative infinity, a negative zero, and many Not-a-Number (NaN) states. The IEEE infinities are said to have the behaviour of limits. Transreal arithmetic is total. It also has a positive and a negative infinity but no negative zero, and it has a single, unordered number, nullity. We elucidate the transreal tangent and extend real limits to transreal limits. Arguing from this firm foundation, we maintain that there are three category errors in the IEEE 754 standard. Firstly the claim that IEEE infinities are limits of real arithmetic confuses limiting processes with arithmetic. Secondly a defence of IEEE negative zero confuses the limit of a function with the value of a function. Thirdly the definition of IEEE NaNs confuses undefined with unordered. Furthermore we prove that the tangent function, with the infinities given by geometrical con- struction, has a period of an entire rotation, not half a rotation as is commonly understood. This illustrates a category error, confusing the limit with the value of a function, in an important area of applied mathe- matics { trigonometry. We brie y consider the wider implications of this category error. Another paper proposes transreal arithmetic as a basis for floating- point arithmetic; here we take the profound step of proposing transreal arithmetic as a replacement for real arithmetic to remove the possibility of certain category errors in mathematics. Thus we propose both theo- retical and practical advantages of transmathematics. In particular we argue that implementing transreal analysis in trans- floating-point arith- metic would extend the coverage, accuracy and reliability of almost all computer programs that exploit real analysis { essentially all programs in science and engineering and many in finance, medicine and other socially beneficial applications.

Estimate of the cutoff errors in the Ewald summation for dipolar systems

Theoretical estimates for the cutoff errors in the Ewald summation method for dipolar systems are derived. Absolute errors in the total energy, forces and torques, both for the real and reciprocal space parts, are considered. The applicability of the estimates is tested and confirmed in several numerical examples. We demonstrate that these estimates can be used easily in determining the optimal parameters of the dipolar Ewald summation in the sense that they minimize the computation time for a predefined, user set, accuracy.

Representation errors and retrievals in linear and nonlinear data assimilation

This article shows how one can formulate the representation problem starting from Bayes’ theorem. The purpose of this article is to raise awareness of the formal solutions,so that approximations can be placed in a proper context. The representation errors appear in the likelihood, and the different possibilities for the representation of reality in model and observations are discussed, including nonlinear representation probability density functions. Specifically, the assumptions needed in the usual procedure to add a representation error covariance to the error covariance of the observations are discussed,and it is shown that, when several sub-grid observations are present, their mean still has a representation error ; socalled ‘superobbing’ does not resolve the issue. Connection is made to the off-line or on-line retrieval problem, providing a new simple proof of the equivalence of assimilating linear retrievals and original observations. Furthermore, it is shown how nonlinear retrievals can be assimilated without loss of information. Finally we discuss how errors in the observation operator model can be treated consistently in the Bayesian framework, connecting to previous work in this area.

Intra-tumor heterogeneity: lessons from microbial evolution and clinical implications

Multiple subclonal populations of tumor cells can coexist within the same tumor. This intra-tumor heterogeneity will have clinical implications and it is therefore important to identify factors that drive or suppress such heterogeneous tumor progression. Evolutionary biology can provide important insights into this process. In particular, experimental evolution studies of microbial populations, which exist as clonal populations that can diversify into multiple subclones, have revealed important evolutionary processes driving heterogeneity within a population. There are transferrable lessons that can be learnt from these studies that will help us to understand the process of intra-tumor heterogeneity in the clinical setting. In this review, we summarize drivers of microbial diversity that have been identified, such as mutation rate and environmental influences, and discuss how knowledge gained from microbial experimental evolution studies may guide us to identify and understand important selective factors that promote intra-tumor heterogeneity. Furthermore, we discuss how these factors could be used to direct and optimize research efforts to improve patient care, focusing on therapeutic resistance. Finally, we emphasize the need for longitudinal studies to address the impact of these potential tumor heterogeneity-promoting factors on drug resistance, metastatic potential and clinical outcome.

Propagation of the Madden–Julian Oscillation and scale interaction with the diurnal cycle in a high-resolution GCM

The Madden–Julian Oscillation (MJO) is the chief source of tropical intra-seasonal variability, but is simulated poorly by most state-of-the-art GCMs. Common errors include a lack of eastward propagation at the correct frequency and zonal extent, and too small a ratio of eastward- to westward-propagating variability. Here it is shown that HiGEM, a high-resolution GCM, simulates a very realistic MJO with approximately the correct spatial and temporal scale. Many MJO studies in GCMs are limited to diagnostics which average over a latitude band around the equator, allowing an analysis of the MJO’s structure in time and longitude only. In this study a wider range of diagnostics is applied. It is argued that such an approach is necessary for a comprehensive analysis of a model’s MJO. The standard analysis of Wheeler and Hendon (Mon Wea Rev 132(8):1917–1932, 2004; WH04) is applied to produce composites, which show a realistic spatial structure in the MJO envelopes but for the timing of the peak precipitation in the inter-tropical convergence zone, which bifurcates the MJO signal. Further diagnostics are developed to analyse the MJO’s episodic nature and the “MJO inertia” (the tendency to remain in the same WH04 phase from one day to the next). HiGEM favours phases 2, 3, 6 and 7; has too much MJO inertia; and dies out too frequently in phase 3. Recent research has shown that a key feature of the MJO is its interaction with the diurnal cycle over the Maritime Continent. This interaction is present in HiGEM but is unrealistically weak.

Finite sample weighting of recursive forecast errors

This paper proposes and tests a new framework for weighting recursive out-of-sample prediction errors according to their corresponding levels of in-sample estimation uncertainty. In essence, we show how to use the maximum possible amount of information from the sample in the evaluation of the prediction accuracy, by commencing the forecasts at the earliest opportunity and weighting the prediction errors. Via a Monte Carlo study, we demonstrate that the proposed framework selects the correct model from a set of candidate models considerably more often than the existing standard approach when only a small sample is available. We also show that the proposed weighting approaches result in tests of equal predictive accuracy that have much better sizes than the standard approach. An application to an exchange rate dataset highlights relevant differences in the results of tests of predictive accuracy based on the standard approach versus the framework proposed in this paper.

Comparison of bioactivities, binding properties and intra-follicular levels of bovine follistatins

Five isoforms of follistatin (FST) (Mr 31, 33, 35, 37, 41kDa) were purified from bovine follicular fluid (bFF). Comparison of their activin- and heparan sulphate proteoglycan (HSP)-binding properties and bio-potencies in neutralization of activin-A action in vitro revealed that all five isoforms bound activin-A, but with different affinities. Only the 31kDa isoform (FST-288) bound to HSP. FST-288 also showed the greatest biopotency with 35 and 41kDa isoforms being least potent. To determine whether bovine follicle development is associated with changing intrafollicular FST and activin profiles, we analyzed bFF from dominant (DF) and subordinate (SF) follicles collected at strategic times during a synchronized estrous cycle. Total FST, activin-A and activin-AB were measured by immunoassay while individual FST isoforms were quantified by immunoblotting. Follicle diameter was positively correlated with estrogen:progesterone ratio (r=0.56) in bFF but negatively correlated with activin-A (r=-0.34), activin-AB (r=-0.80) and ‘total’ FST (r=-0.70) levels. Follicle diameter was positively correlated with abundance of the 41 kDa isoform (r=0.59) but negatively correlated with abundance of 33 and 31 kDa isoforms (r=-0.56, -0.41). Both follicle status (DF vs SF) and cycle stage affected total FST, activin-A, activin-B levels while follicle status, but not cycle stage, affected abundance of 41, 37, 33 and 31kDa FST isoforms. Collectively, these findings indicate that intrafollicular FST isoforms that differ in their ability to bind and neutralise activins and associate with cell-surface proteoglycans, show divergent changes during follicle development. Enhanced FST production may have an important negative role, either directly or via inhibition of the positive effects of activins, on follicle growth and function during follicular waves.

Stabilization of the i-motif structure by the intra-strand cross-link formation

The i-motif structures are formed by oligonucleotides containing cytosine tracts under acidic conditions. The folding of the i-motif under physiological conditions is of great interest because of its biological role. In this study, we investigated the effect of the intra-strand cross-link on the stability of the i-motif structure. The 4-vinyl-substituted analog of thymidine (T-vinyl) was incorporated into the 5′-end of the human telomere complementary strand, which formed the intra-strand cross-link with the internal adenine. The intra-strand cross-linked i-motif displayed CD spectra similar to that of the natural i-motif at acidic pH, which was transformed into a random coil with the increasing pH. The pH midpoint for the transition from the i-motif to random coil increased from pH 6.1 for the natural one to pH 6.8 for the cross-linked one. The thermodynamic parameters were obtained by measuring the thermal melting behaviors by CD and UV, and it was determined that the intra-strand cross-linked i-motif is stabilized due to a favorable entropy effect. Thus, this study has clearly indicated the validity of the intra-strand cross-linking for stabilization of the i-motif structure.

Pragmatic solutions for syntactic problems: Understanding some L2 syntactic errors in terms of discourse-pragmatic deficits

Contemporary research in generative second language (L2) acquisition has attempted to address observable target-deviant aspects of L2 grammars within a UG-continuity framework (e.g. Lardiere 2000; Schwartz 2003; Sprouse 2004; Prévost & White 1999, 2000). With the aforementioned in mind, the independence of pragmatic and syntactic development, independently observed elsewhere (e.g. Grodzinsky & Reinhart 1993; Lust et al. 1986; Pacheco & Flynn 2005; Serratrice, Sorace & Paoli 2004), becomes particularly interesting. In what follows, I examine the resetting of the Null-Subject Parameter (NSP) for English learners of L2 Spanish. I argue that insensitivity to associated discoursepragmatic constraints on the discursive distribution of overt/null subjects accounts for what appear to be particular errors as a result of syntactic deficits. It is demonstrated that despite target-deviant performance, the majority must have native-like syntactic competence given their knowledge of the Overt Pronoun Constraint (Montalbetti 1984), a principle associated with the Spanish-type setting of the NSP.

Tests of sunspot number sequences: 3. Effects of regression procedures on the calibration of historic sunspot data

We use sunspot group observations from the Royal Greenwich Observatory (RGO) to investigate the effects of intercalibrating data from observers with different visual acuities. The tests are made by counting the number of groups RB above a variable cut-off threshold of observed total whole-spot area (uncorrected for foreshortening) to simulate what a lower acuity observer would have seen. The synthesised annual means of RB are then re-scaled to the full observed RGO group number RA using a variety of regression techniques. It is found that a very high correlation between RA and RB (rAB > 0.98) does not prevent large errors in the intercalibration (for example sunspot maximum values can be over 30 % too large even for such levels of rAB). In generating the backbone sunspot number (RBB), Svalgaard and Schatten (2015, this issue) force regression fits to pass through the scatter plot origin which generates unreliable fits (the residuals do not form a normal distribution) and causes sunspot cycle amplitudes to be exaggerated in the intercalibrated data. It is demonstrated that the use of Quantile-Quantile (“Q  Q”) plots to test for a normal distribution is a useful indicator of erroneous and misleading regression fits. Ordinary least squares linear fits, not forced to pass through the origin, are sometimes reliable (although the optimum method used is shown to be different when matching peak and average sunspot group numbers). However, other fits are only reliable if non-linear regression is used. From these results it is entirely possible that the inflation of solar cycle amplitudes in the backbone group sunspot number as one goes back in time, relative to related solar-terrestrial parameters, is entirely caused by the use of inappropriate and non-robust regression techniques to calibrate the sunspot data.

Systematic and random errors between collocated satellite ice water path observations

There remains large disagreement between ice-water path (IWP) in observational data sets, largely because the sensors observe different parts of the ice particle size distribution. A detailed comparison of retrieved IWP from satellite observations in the Tropics (!30 " latitude) in 2007 was made using collocated measurements. The radio detection and ranging(radar)/light detection and ranging (lidar) (DARDAR) IWP data set, based on combined radar/lidar measurements, is used as a reference because it provides arguably the best estimate of the total column IWP. For each data set, usable IWP dynamic ranges are inferred from this comparison. IWP retrievals based on solar reflectance measurements, in the moderate resolution imaging spectroradiometer (MODIS), advanced very high resolution radiometer–based Climate Monitoring Satellite Applications Facility (CMSAF), and Pathfinder Atmospheres-Extended (PATMOS-x) datasets, were found to be correlated with DARDAR over a large IWP range (~20–7000 g m -2 ). The random errors of the collocated data sets have a close to lognormal distribution, and the combined random error of MODIS and DARDAR is less than a factor of 2, which also sets the upper limit for MODIS alone. In the same way, the upper limit for the random error of all considered data sets is determined. Data sets based on passive microwave measurements, microwave surface and precipitation products system (MSPPS), microwave integrated retrieval system (MiRS), and collocated microwave only (CMO), are largely correlated with DARDAR for IWP values larger than approximately 700 g m -2 . The combined uncertainty between these data sets and DARDAR in this range is slightly less MODIS-DARDAR, but the systematic bias is nearly an order of magnitude.

High-precision measurements of the co-polar correlation coefficient: non-Gaussian errors and retrieval of the dispersion parameter µ in rainfall

The co-polar correlation coefficient (ρhv) has many applications, including hydrometeor classification, ground clutter and melting layer identification, interpretation of ice microphysics and the retrieval of rain drop size distributions (DSDs). However, we currently lack the quantitative error estimates that are necessary if these applications are to be fully exploited. Previous error estimates of ρhv rely on knowledge of the unknown "true" ρhv and implicitly assume a Gaussian probability distribution function of ρhv samples. We show that frequency distributions of ρhv estimates are in fact highly negatively skewed. A new variable: L = -log10(1 - ρhv) is defined, which does have Gaussian error statistics, and a standard deviation depending only on the number of independent radar pulses. This is verified using observations of spherical drizzle drops, allowing, for the first time, the construction of rigorous confidence intervals in estimates of ρhv. In addition, we demonstrate how the imperfect co-location of the horizontal and vertical polarisation sample volumes may be accounted for. The possibility of using L to estimate the dispersion parameter (µ) in the gamma drop size distribution is investigated. We find that including drop oscillations is essential for this application, otherwise there could be biases in retrieved µ of up to ~8. Preliminary results in rainfall are presented. In a convective rain case study, our estimates show µ to be substantially larger than 0 (an exponential DSD). In this particular rain event, rain rate would be overestimated by up to 50% if a simple exponential DSD is assumed.

A moving observer in a three-dimensional world

For many tasks, such as retrieving a previously viewed object, an observer must form a representation of the world at one location and use it at another. A world-based 3D reconstruction of the scene built up from visual information would fulfil this requirement, something computer vision now achieves with great speed and accuracy. However, I argue that it is neither easy nor necessary for the brain to do this. I discuss biologically plausible alternatives, including the possibility of avoiding 3D coordinate frames such as ego-centric and world-based representations. For example, the distance, slant and local shape of surfaces dictate the propensity of visual features to move in the image with respect to one another as the observer’s perspective changes (through movement or binocular viewing). Such propensities can be stored without the need for 3D reference frames. The problem of representing a stable scene in the face of continual head and eye movements is an appropriate starting place for understanding the goal of 3D vision, more so, I argue, than the case of a static binocular observer.

Recurrent sores by ill-fitting dentures and intra-oral squamous cell carcinoma in smokers

Objectives: To examine whether denture use and recurrent sores caused by ill-fitting dentures are associated with intra-oral squamous cell carcinoma (IO-SCC) in individuals exposed to tobacco. Methods: We conducted a hospital-based case-control study. The study population comprised 124 patients with IO-SCC and the same number of controls (individually paired according to gender and age) recruited from outpatient units of the same hospital. Conditional logistic regression analysis assessed the effect of denture use and recurrent oral sores by ill-fitting dentures, adjusted by covariates on the lifetime exposure to alcohol and tobacco, socioeconomic standings, and dietary patterns. Results: The use of dentures showed no association with IO-SCC [adjusted odds ratio (OR) 1.40, 95 percent confidence interval 0.51-3.87, P = 0.513] in an assessment controlled by socioeconomic position, lifetime exposure to alcohol and tobacco, and dietary patterns. However, the report of recurrent sores caused by ill-fitting dentures showed significant association with the disease (adjusted OR 4.58, 95 percent confidence interval 1.52-13.76, P = 0.007). Conclusions: The association between recurrent oral sores caused by ill-fitting dentures and squamous cell carcinoma of the mouth in smokers is in agreement with the hypothesis that the chronic physical irritation of oral mucosa contributes to the topical carcinogenic effect of tobacco, which must be taken into careful consideration in the planning of dental services for adults and the elderly.

Intra-strain variations of baroreflex sensitivity in young Wistar-Kyoto rats

Purpose: To compare baroreflex sensitivity among conscious rats of the same strain. Methods: Male WKY rats (eight weeks old) were studied. Cannulas were inserted into the abdominal aortic artery through the right femoral artery to measure mean arterial pressure (MAP) and heart rate (HR). Baroreflex gain was calculated as the ratio between variation of HR in function of the MAP variation (Delta HR/Delta MAP) tested with a depressor dose of sodium nitroprusside (SNP, 50 mu g/kg, iv) and with a pressor dose of phenylephrine (PE, 8 mu g/kg, iv). We divided the rats into four groups: 1) Low bradycardic baroreflex (LB), BG between -1 and -2 bpm/mmHg tested with PE; 2) High bradycardic baroreflex (HB), BG < -2 bpm/mmHg tested with PE; 3) Low tachycardic baroreflex (LT), BG between -1 and -2 bpm/mmHg tested with SNP and; 4) High tachycardic baroreflex (HT), BG < -2 bpm/mmHg tested with SNP. Significant differences were considered for p<0.05. Results: Approximately 82% of the rats presented reduced bradycardic reflex while 22 showed attenuated tachycardic reflex. No alterations were noted regarding basal MAP and HR, tachycardic and bradycardic peak and HR range. Conclusions: There was alteration in baroreflex sensitivity among rats of the same strain. Care should be taken when interpreting studies employing WKY as a control for the SHR.