On the size of the fibers of spectral maps induced by semialgebraic embeddings

Let S(M) be the ring of (continuous) semialgebraic functions on a semialgebraic set M and S*(M) its subring of bounded semialgebraic functions. In this work we compute the size of the fibers of the spectral maps Spec(j)1:Spec(S(N))→Spec(S(M)) and Spec(j)2:Spec(S*(N))→Spec(S*(M)) induced by the inclusion j:N M of a semialgebraic subset N of M. The ring S(M) can be understood as the localization of S*(M) at the multiplicative subset WM of those bounded semialgebraic functions on M with empty zero set. This provides a natural inclusion iM:Spec(S(M)) Spec(S*(M)) that reduces both problems above to an analysis of the fibers of the spectral map Spec(j)2:Spec(S*(N))→Spec(S*(M)). If we denote Z:=ClSpec(S*(M))(M N), it holds that the restriction map Spec(j)2|:Spec(S*(N)) Spec(j)2-1(Z)→Spec(S*(M)) Z is a homeomorphism. Our problem concentrates on the computation of the size of the fibers of Spec(j)2 at the points of Z. The size of the fibers of prime ideals "close" to the complement Y:=M N provides valuable information concerning how N is immersed inside M. If N is dense in M, the map Spec(j)2 is surjective and the generic fiber of a prime ideal p∈Z contains infinitely many elements. However, finite fibers may also appear and we provide a criterium to decide when the fiber Spec(j)2-1(p) is a finite set for p∈Z. If such is the case, our procedure allows us to compute the size s of Spec(j)2-1(p). If in addition N is locally compact and M is pure dimensional, s coincides with the number of minimal prime ideals contained in p. © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Tear Secretion Induced by Selective Stimulation of Corneal and Conjunctival Sensory Nerve Fibers

Purpose. To measure the increase in tear secretion evoked by selective stimulation of the different populations of sensory receptors of the cornea and conjunctiva by using moderate and intense mechanical, chemical, and cold stimuli. Methods. Six healthy subjects participated in the study. Tear secretion was measured in both eyes by the Schirmer’s test conducted under control conditions and after stimulation of the center of the cornea and the temporal conjunctiva with a gas esthesiometer. Mechanical stimulation consisted in three pulses of 3 seconds’ duration of warmed air (at 34°C on the eye surface) applied at moderate (170 mL/min) and high (260 mL/min) flow rates. Cold thermal stimulation was made with cooled air that produced a corneal temperature drop of −1°C or −4.5°C. Chemical (acidic) stimulation was performed with a jet of gas containing a mixture of 80% CO2 in air. Results. The basal volume of tear secretion increased significantly (P < 0.05, paired t-test) after stimulation of the cornea with high-flow mechanical stimuli (260 mL/min), intense cooling pulses (−4.5°C), and chemical stimulation (80% CO2). The same stimuli were ineffective when applied to the conjunctiva. Moderate mechanical (170 mL/min) and cold (−1°C) stimulation of the cornea or the conjunctiva did not change significantly the volume of tear secretion. Conclusions. Reflex tear secretion caused by corneal stimulation seems to be chiefly due to activation of corneal polymodal nociceptors, whereas selective excitation of corneal mechanonociceptors or cold receptors appears to be less effective in evoking an augmented lacrimal secretion. Conjunctival receptors stimulated at equivalent levels do not evoke an increased tear secretion.