The Minimum Number of Points Taking Part in k-Sets in Sets of Unaligned Points

En este trabajo se da un ejemplo de un conjunto de n puntos situados en posición general, en el que se alcanza el mínimo número de puntos que pueden formar parte de algún k-set para todo k con 1menor que=kmenor quen/2. Se generaliza también, a puntos en posición no general, el resultado de Erdõs et al., 1973, sobre el mínimo número de puntos que pueden formar parte de algún k-set. The study of k- sets is a very relevant topic in the research area of computational geometry. The study of the maximum and minimum number of k-sets in sets of points of the plane in general position, specifically, has been developed at great length in the literature. With respect to the maximum number of k-sets, lower bounds for this maximum have been provided by Erdõs et al., Edelsbrunner and Welzl, and later by Toth. Dey also stated an upper bound for this maximum number of k-sets. With respect to the minimum number of k-set, this has been stated by Erdos el al. and, independently, by Lovasz et al. In this paper the authors give an example of a set of n points in the plane in general position (no three collinear), in which the minimum number of points that can take part in, at least, a k-set is attained for every k with 1 ≤ k < n/2. The authors also extend Erdos’s result about the minimum number of points in general position which can take part in a k-set to a set of n points not necessarily in general position. That is why this work complements the classic works we have mentioned before.

Defective γ-aminobutyric acid type B receptor-activated inwardly rectifying K+ currents in cerebellar granule cells isolated from weaver and Girk2 null mutant mice

Stimulation of inhibitory neurotransmitter receptors, such as γ-aminobutyric acid type B (GABAB) receptors, activates G protein-gated inwardly rectifying K+ channels (GIRK) which, in turn, influence membrane excitability. Seizure activity has been reported in a Girk2 null mutant mouse lacking GIRK2 channels but showing normal cerebellar development as well as in the weaver mouse, which has mutated GIRK2 channels and shows abnormal development. To understand how the function of GIRK2 channels differs in these two mutant mice, we compared the G protein-activated inwardly rectifying K+ currents in cerebellar granule cells isolated from Girk2 null mutant and weaver mutant mice with those from wild-type mice. Activation of GABAB receptors in wild-type granule cells induced an inwardly rectifying K+ current, which was sensitive to pertussis toxin and inhibited by external Ba2+ ions. The amplitude of the GABAB receptor-activated current was severely attenuated in granule cells isolated from both weaver and Girk2 null mutant mice. By contrast, the G protein-gated inwardly rectifying current and possibly the agonist-independent basal current appeared to be less selective for K+ ions in weaver but not Girk2 null mutant granule cells. Our results support the hypothesis that a nonselective current leads to the weaver phenotype. The loss of GABAB receptor-activated GIRK current appears coincident with the absence of GIRK2 channel protein and the reduction of GIRK1 channel protein in the Girk2 null mutant mouse, suggesting that GABAB receptors couple to heteromultimers composed of GIRK1 and GIRK2 channel subunits.

β-Neuregulin-1 is required for the in vivo development of functional Ca2+-activated K+ channels in parasympathetic neurons

The development of functional Ca2+-activated K+ channels (KCa) in chick ciliary ganglion (CG) neurons requires interactions with afferent preganglionic nerve terminals. Here we show that the essential preganglionic differentiation factor is an isoform of β-neuregulin-1. β-Neuregulin-1 transcripts are expressed in the midbrain preganglionic Edinger–Westphal nucleus at developmental stages that coincide with or precede the normal onset of macroscopic KCa in CG neurons. Injection of β-neuregulin-1 peptide into the brains of developing embryos evoked a robust stimulation of functional KCa channels at stages before the normal appearance of these channels in CG neurons developing in vivo. Conversely, injection of a neutralizing antiserum specific for β-neuregulin-1 inhibited the development of KCa channels in CG neurons. Low concentrations of β-neuregulin-1 evoked a robust increase in whole-cell KCa in CG neurons cocultured with iris target tissues. By contrast, culturing CG neurons with iris cells or low concentrations of β-neuregulin-1 by themselves was insufficient to stimulate KCa. These data suggest that the preganglionic factor required for the development of KCa in ciliary ganglion neurons is an isoform of β-neuregulin-1, and that this factor acts in concert with target-derived trophic molecules to regulate the differentiation of excitability.

Regulation of K+ Channels in Maize Roots by Water Stress and Abscisic Acid1

Root cortical and stelar protoplasts were isolated from maize (Zea mays L.) plants that were either well watered or water stressed, and the patch-clamp technique was used to investigate their plasma membrane K+ channel activity. In the root cortex water stress did not significantly affect inward- or outward-rectifying K+ conductances relative to those observed in well-watered plants. In contrast, water stress significantly reduced the magnitude of the outward-rectifying K+ current in the root stele but had little effect on the inward-rectifying K+ current. Pretreating well-watered plants with abscisic acid also significantly affected K+ currents in a way that was consistent with abscisic acid mediating, at least in part, the response of roots to water stress. It is proposed that the K+ channels underlying the K+ currents in the root stelar cells represent pathways that allow K+ exchange between the root symplasm and xylem apoplast. It is suggested that the regulation of K+ channel activity in the root in response to water stress could be part of an important adaptation of the plant to survive drying soils.

The weaver mutation of GIRK2 results in a loss of inwardly rectifying K+ current in cerebellar granule cells.

The weaver mutation in mice results in a severe ataxia that is attributable to the degeneration of cerebellar granule cells and dopaminergic neurons in the substantia nigra. Recent genetic studies indicate that the GIRK2 gene is altered in weaver. This gene codes for a G-protein-activated, inwardly rectifying K+ channel protein (8). The mutation results in a single amino acid substitution (glycine-->serine) in the pore-forming H5 region of the channel. The functional consequences of this mutation appear to depend upon the co-expression of other GIRK subunits--leading to either a gain or loss of function. Here, we show that G-protein-activated inwardly rectifying K+ currents are significantly reduced in cerebellar granule cells from animals carrying the mutant allele. The reduction is most pronounced in homozygous neurons. These findings suggest that the death of neurons in weaver is attributable to the loss of GIRK2-mediated currents, not to the expression of a nonspecific cation current.

cAMP-dependent, long-lasting inhibition of a K+ current in mammalian neurons.

We report the long-term modulation of K+ channels by cAMP in cultured murine colliculi neurons. A short (1-2 s) application of 8-Br-cAMP induced a long-lasting broadening of the action potential, a loss of after-hyperpolarization, and a reduction in spike accommodation. In agreement with these changes, 8-Br-cAMP produced a long-lasting (2 hr) inhibition of a K+ current. These effects were also observed after a short activation of the pituitary adenylyl cyclase-activating polypeptide, beta-adrenergic, and 5-hydroxytryptamine type 4 (5-HT4) receptors, all known to increase cAMP. A transient activation of the cAMP-dependent protein kinase and a long-lasting inhibition of phosphatases (up to 2 hr) were detected. The blockade of the K+ current resulting from a brief application of 8-Br-cAMP or 5-hydroxytryptamine was prolonged from 2 to 4 hr when protein-serine/threonine phosphatases 1 and 2A were inhibited with 10 nM okadaic acid. The critical steps following the cAMP-dependent protein kinase activation and resulting in a long-term blockade of phosphatases are discussed in this report.

Essential role of adenosine, adenosine A1 receptors, and ATP-sensitive K+ channels in cerebral ischemic preconditioning.

Preconditioning with sublethal ischemia protects against neuronal damage after subsequent lethal ischemic insults in hippocampal neurons. A pharmacological approach using agonists and antagonists at the adenosine A1 receptor as well as openers and blockers of ATP-sensitive K+ channels has been combined with an analysis of neuronal death and gene expression of subunits of glutamate and gamma-aminobutyric acid receptors, HSP70, c-fos, c-jun, and growth factors. It indicates that the mechanism of ischemic tolerance involves a cascade of events including liberation of adenosine, stimulation of adenosine A1 receptors, and, via these receptors, opening of sulfonylurea-sensitive ATP-sensitive K+ channels.

Di ḳoḳeṭe damen : ḳomedye in fier aḳṭen miṭ a prolog.

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Die Entwicklung der K.K. Landwirtschaftsgesellschaft in Wien während ihres hundertjährigen Bestandes /

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Aytsʻ mě Ani kʻaghakʻin /

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Ihr stilles gluc̈k--! Drama in fünf Aufzügen

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Chanaparhordutʻiwn i Lehastan ew yayl koghmans bnakeals i haykaghantsʻ serelotsʻ i nakhneatsʻ Ani kʻaghakʻin : sharagreal handerdz zanazan banasirakan teghekutʻeamb /

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Speech of the Marquess of Anglesey, K.G. : in the House of Lords : on Monday, the 4th of May, 1829, on moving for papers relative to his recall from the government of Ireland.

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Altdeutsche Gedichte aus den Zeiten der Tafelrunde. Aus handschriften der k. k. Hofbibliothek in die heutige Sprache übertragen.

Engr. title pages; v. 1 has added t. p., printed.