Based on probability theory, a methodology that allows diagnosing neonatal cardiac dynamics was previously developed; however, diagnostic applications of this method are required to validate it to the neonatal cardiac dynamics was conducted, allowing to differentiate normal from pathological dynamics. The hourly maximum and minimum heart rate values from 39 continuous and ambulatory electrocardiographic records with a minimum length of 21 hours were taken, from newborns between 0 and 10 days of life, 9 clinically within normality limits and 30 with cardiac pathologies. The probability of occurrence of heart rates in ranges of 5 beats/minute was calculated. The distributions of probability were analysed, and finally the diagnosis was determined by the physical-mathematical methodology. Then, a statistical validation of sensitivity, specificity, and diagnostic agreement was performed. Normal registries showed probability distributions with absent or minimal presence of heart rates of the ranges between 125 and 135 beats/minute, while the abnormal ones had values within these ranges, as well as absence or minimal presence of heart rates from 75 beats/minute to 85 beats/minute. The sensitivity and specificity were 100%, and the Kappa coefficient had a value of 1. Hereby, it is concluded that through an application of a physical–mathematical methodology of neonatal cardiac diagnosis, it is possible to differentiate normality from disease.

A Cantor series expansion for a real number x with respect to a basic sequence $Q=(q_1,q_2,\dots )$ , where $q_i \geq 2$ , is a generalization of the base b expansion to an infinite sequence of bases. Ki and Linton in 1994 showed that for ordinary base b expansions the set of normal numbers is a $\boldsymbol {\Pi }^0_3$ -complete set, establishing the exact complexity of this set. In the case of Cantor series there are three natural notions of normality: normality, ratio normality, and distribution normality. These notions are equivalent for base b expansions, but not for more general Cantor series expansions. We show that for any basic sequence the set of distribution normal numbers is $\boldsymbol {\Pi }^0_3$ -complete, and if Q is $1$ -divergent then the sets of normal and ratio normal numbers are $\boldsymbol {\Pi }^0_3$ -complete. We further show that all five non-trivial differences of these sets are $D_2(\boldsymbol {\Pi }^0_3)$ -complete if $\lim _i q_i=\infty $ and Q is $1$ -divergent. This shows that except for the trivial containment that every normal number is ratio normal, these three notions are as independent as possible.

We prove that digital sequences modulo $m$ along squares are normal, which covers some prominent sequences, such as the sum of digits in base $q$ modulo $m$, the Rudin–Shapiro sequence, and some generalizations. This gives, for any base, a class of explicit normal numbers that can be efficiently generated.

The arcsine and square root transformations were tested on 82 weed control data sets and 62 winter wheat winter survival data sets to determine effects on normality of the error terms, homogeneity of variance, and additivity of the model. Transformations appeared to correct deficiencies in these three parameters in the majority of data sets, but had adverse effects in certain other data sets. Performing the recommended transformation in conjunction with omitting treatments having identical replicate observations provided a high percentage of correction of non-normality, heterogeneity of variance, and nonadditivity. The arcsine transformation, not generally recommended for data sets having values from 0 to 20% or 80 to 100%, was as effective in correcting non-normality, heterogeneity of variance, and nonadditivity in these data sets as was the recommended square root transformation. A majority of data sets showed differences between transformed and nontransformed data in mean separations determined using LSD (0.05), although most of these differences were minor and had little effect on interpretation of results.

According to human enhancement advocates, it is morally permissible (and sometimes obligatory) to use biomedical means to modulate or select certain biological traits in order to increase people’s welfare, even when there is no pathology to be treated or prevented. Some authors have recently proposed to extend the use of biomedical means to modulate lust, attraction, and attachment. I focus on some conceptual implications of this proposal, particularly with regard to bioconservatives’ understanding of the notions of therapy and enhancement I first explain what makes the proposal of medicalizing love interesting and unique, compared to the other forms of bioenhancement usually advocated. I then discuss how the medicalization of love bears on the more general debate on human enhancement, particularly with regard to the key notion of “normality” that is commonly used to define the therapy–enhancement distinction. This analysis suggests that the medicalization of love, in virtue of its peculiarity, requires bioconservatives to reconsider their way of understanding and applying the notions of “therapy” and “enhancement.” More in particular, I show that, because a non-arbitrary and value-free notion of “therapy” cannot be applied to the case of love, bioconservatives have the burden of either providing some new criterion that could be used for drawing a line between permissible and impermissible medicalization, or demonstrating that under no circumstances—including the cases in which love is already acknowledged to require medical intervention—can love fall within the domain of medicine.

Recent empirical findings suggest that macroeconomic variables are seldom normally distributed. For example, the distributions of aggregate output growth-rate time series of many OECD countries are well approximated by symmetric exponential-power (EP) densities with Laplace fat tails. In this work, we assess whether real business cycle (RBC) and standard medium-scale New Keynesian (NK) models are able to replicate this statistical regularity. We simulate both models, drawing Gaussian- vs Laplace-distributed shocks, and we explore the statistical properties of simulated time series. Our results cast doubts on whether RBC and NK models are able to provide a satisfactory representation of the transmission mechanisms linking exogenous shocks to macroeconomic dynamics.

Let ℱ be a family of zero-free meromorphic functions in a domain D, let h be a holomorphic function in D, and let k be a positive integer. If the function f(k)−h has at most k distinct zeros (ignoring multiplicity) in D for each f∈ℱ, then ℱ is normal in D.

In this paper we consider linear Hamiltonian differential systems without the controllability (or normality) assumption. We prove the Rayleigh principle for these systems with Dirichlet boundary conditions, which provides a variational characterization of the finite eigenvalues of the associated self-adjoint eigenvalue problem. This result generalizes the traditional Rayleigh principle to possibly abnormal linear Hamiltonian systems. The main tools are the extended Picone formula, which is proven here for this general setting, results on piecewise constant kernels for conjoined bases of the Hamiltonian system, and the oscillation theorem relating the number of proper focal points of conjoined bases with the number of finite eigenvalues. As applications we obtain the expansion theorem in the space of admissible functions without controllability and a result on coercivity of the corresponding quadratic functional.

Let ℱ be a family of meromorphic functions defined in D, all of whose zeros have multiplicity at least k+1. Let a and b be distinct finite complex numbers, and let k be a positive integer. If, for each pair of functions f and g in ℱ, f(k) and g(k) share the set S={a,b}, then ℱ is normal in D. The condition that the zeros of functions in ℱ have multiplicity at least k+1 cannot be weakened.

Constants with formulae of the form treated by D. Bailey, P. Borwein, and S. Plouffe ($b$ formulae to a given base $b$) have interesting computational properties, such as allowing single digits in their base $b$ expansion to be independently computed, and there are hints that they should be normal numbers, i.e., that their base $b$ digits are randomly distributed. We study a formally limited subset of BBP formulae, which we call Machin-type BBP formulae, for which it is relatively easy to determine whether or not a given constant $K$ has a Machin-type BBP formula. In particular, given $b\,\in \,\mathbb{N},\,b\,>\,2,\,b$ not a proper power, a $b$-ary Machin-type BBP arctangent formula for $K$ is a formula of the form $k\,=\,{{\Sigma }_{m}}\,{{a}_{m}}\,\arctan \,(-{{b}^{-m}}),\,{{a}_{m}}\,\in \,\mathbb{Q}$ , while when $b\,=\,2$, we also allow terms of the form ${{a}_{m}}\,\arctan \,(1/1\,-\,{{2}^{m}}))$ . Of particular interest, we show that $\pi$ has no Machin-type BBP arctangent formula when $b\,\ne \,2$. To the best of our knowledge, when there is no Machin-type BBP formula for a constant then no BBP formula of any form is known for that constant.

Let k be a positive integer and b a nonzero constant. Suppose that F is a family of meromorphic functions in a domain D. If each function f ∈ F has only zeros of multiplicity at least k + 2 and for any two functions f, g ∈ F, f and g share 0 in D and f(k) and g(k) share b in D, then F is normal in D. The case f ≠ 0, f(k) ≠ b is a celebrated result of Gu.

We shall discuss relations between rectangularity and piecewise rectangularity of product spaces. In particular, we show that for each positive integer n there exists an n-dimensional, collectionwise normal, non-piecewise rectangular product X × Y which satisfies the inequality dim (X × Y) ≤ dim X + dim Y.