A singular function and its relation with the number systems involved in its definition

Minkowski's ?(x) function can be seen as the confrontation of two number systems: regular continued fractions and the alternated dyadic system. This way of looking at it permits us to prove that its derivative, as it also happens for many other non-decreasing singular functions from [0,1] to [0,1], when it exists can only attain two values: zero and infinity. It is also proved that if the average of the partial quotients in the continued fraction expansion of x is greater than k* =5.31972, and ?'(x) exists then ?'(x)=0. In the same way, if the same average is less than k**=2 log2(F), where F is the golden ratio, then ?'(x)=infinity. Finally some results are presented concerning metric properties of continued fraction and alternated dyadic expansions.

Genome-wide association study of major recurrent depression in the U.K. population.

OBJECTIVE: Studies of major depression in twins and families have shown moderate to high heritability, but extensive molecular studies have failed to identify susceptibility genes convincingly. To detect genetic variants contributing to major depression, the authors performed a genome-wide association study using 1,636 cases of depression ascertained in the U.K. and 1,594 comparison subjects screened negative for psychiatric disorders. METHOD: Cases were collected from 1) a case-control study of recurrent depression (the Depression Case Control [DeCC] study; N=1346), 2) an affected sibling pair linkage study of recurrent depression (probands from the Depression Network [DeNT] study; N=332), and 3) a pharmacogenetic study (the Genome-Based Therapeutic Drugs for Depression [GENDEP] study; N=88). Depression cases and comparison subjects were genotyped at Centre National de Génotypage on the Illumina Human610-Quad BeadChip. After applying stringent quality control criteria for missing genotypes, departure from Hardy-Weinberg equilibrium, and low minor allele frequency, the authors tested for association to depression using logistic regression, correcting for population ancestry. RESULTS: Single nucleotide polymorphisms (SNPs) in BICC1 achieved suggestive evidence for association, which strengthened after imputation of ungenotyped markers, and in analysis of female depression cases. A meta-analysis of U.K. data with previously published results from studies in Munich and Lausanne showed some evidence for association near neuroligin 1 (NLGN1) on chromosome 3, but did not support findings at BICC1. CONCLUSIONS: This study identifies several signals for association worthy of further investigation but, as in previous genome-wide studies, suggests that individual gene contributions to depression are likely to have only minor effects, and very large pooled analyses will be required to identify them.

Kenraali K. O. Leinonen 5/10 1973

Kirje 5.10.1973

Presidentti J.K. Paasikivi 27/11 1956

Kirje 27.11.1956

Riesz-Nágy singular functions revisited

In 1952 F. Riesz and Sz.Nágy published an example of a monotonic continuous function whose derivative is zero almost everywhere, that is to say, a singular function. Besides, the function was strictly increasing. Their example was built as the limit of a sequence of deformations of the identity function. As an easy consequence of the definition, the derivative, when it existed and was finite, was found to be zero. In this paper we revisit the Riesz-N´agy family of functions and we relate it to a system for real numberrepresentation which we call (t, t-1) expansions. With the help of these real number expansions we generalize the family. The singularity of the functions is proved through some metrical properties of the expansions used in their definition which also allows us to give a more precise way of determining when the derivative is 0 or infinity.

Effects of mineralocorticoid and K+ concentration on K+ secretion and ROMK channel expression in a mouse cortical collecting duct cell line.

The cortical collecting duct (CCD) plays a key role in regulated K(+) secretion, which is mediated mainly through renal outer medullary K(+) (ROMK) channels located in the apical membrane. However, the mechanisms of the regulation of urinary K(+) excretion with regard to K(+) balance are not well known. We took advantage of a recently established mouse CCD cell line (mCCD(cl1)) to investigate the regulation of K(+) secretion by mineralocorticoid and K(+) concentration. We show that this cell line expresses ROMK mRNA and a barium-sensitive K(+) conductance in its apical membrane. As this conductance is sensitive to tertiapin-Q, with an apparent affinity of 6 nM, and to intracellular acidification, it is probably mediated by ROMK. Overnight exposure to 100 nM aldosterone did not significantly change the K(+) conductance, while it increased the amiloride-sensitive Na(+) transport. Overnight exposure to a high K(+) (7 mM) concentration produced a small but significant increase in the apical membrane barium-sensitive K(+) conductance. The mRNA levels of all ROMK isoforms measured by qRT-PCR were not changed by altering the basolateral K(+) concentration but were decreased by 15-45% upon treatment with aldosterone (0.3 or 300 nM for 1 and 3 h). The paradoxical response of ROMK expression to aldosterone could possibly work as a preventative mechanism to avoid excessive K(+) loss which would otherwise result from the increased electrogenic Na(+) transport and associated depolarization of the apical membrane in the CCD. In conclusion, mCCD(cl1) cells demonstrate a significant K(+) secretion, probably mediated by ROMK, which is not stimulated by aldosterone but increased by overnight exposure to a high K(+) concentration.

Downregulation of cannabinoid receptor 1 from neuropeptide Y interneurons in the basal ganglia of patients with Huntington's disease and mouse models.

Cannabinoid receptor 1 (CB(1) receptor) controls several neuronal functions, including neurotransmitter release, synaptic plasticity, gene expression and neuronal viability. Downregulation of CB(1) expression in the basal ganglia of patients with Huntington's disease (HD) and animal models represents one of the earliest molecular events induced by mutant huntingtin (mHtt). This early disruption of neuronal CB(1) signaling is thought to contribute to HD symptoms and neurodegeneration. Here we determined whether CB(1) downregulation measured in patients with HD and mouse models was ubiquitous or restricted to specific striatal neuronal subpopulations. Using unbiased semi-quantitative immunohistochemistry, we confirmed previous studies showing that CB(1) expression is downregulated in medium spiny neurons of the indirect pathway, and found that CB(1) is also downregulated in neuropeptide Y (NPY)/neuronal nitric oxide synthase (nNOS)-expressing interneurons while remaining unchanged in parvalbumin- and calretinin-expressing interneurons. CB(1) downregulation in striatal NPY/nNOS-expressing interneurons occurs in R6/2 mice, Hdh(Q150/Q150) mice and the caudate nucleus of patients with HD. In R6/2 mice, CB(1) downregulation in NPY/nNOS-expressing interneurons correlates with diffuse expression of mHtt in the soma. This downregulation also occludes the ability of cannabinoid agonists to activate the pro-survival signaling molecule cAMP response element-binding protein in NPY/nNOS-expressing interneurons. Loss of CB(1) signaling in NPY/nNOS-expressing interneurons could contribute to the impairment of basal ganglia functions linked to HD.