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

Let ${\mathcal S}(M)$ be the ring of (continuous) semialgebraic functions on a semialgebraic set $M\subset{\mathbb R}^m$ and ${\mathcal S}^*(M)$ its subring of bounded semialgebraic functions. In this work we compute the size of the fibers of the spectral maps ${\rm Spec}({\tt j})_1:{\rm Spec}({\mathcal S}(N))\to{\rm Spec}({\mathcal S}(M))$ and ${\rm Spec}({\tt j})_2:{\rm Spec}({\mathcal S}^*(N))\to{\rm Spec}({\mathcal S}^*(M))$ induced by the inclusion ${\tt j}:N\hookrightarrow M$ of a semialgebraic subset $N$ of $M$. The ring ${\mathcal S}(M)$ can be understood as the localization of ${\mathcal S}^*(M)$ at the multiplicative subset ${\mathcal W}_M$ of those bounded semialgebraic functions on $M$ with empty zero set. This provides a natural inclusion ${\mathfrak i}_M:{\rm Spec}({\mathcal S}(M))\hookrightarrow{\rm Spec}({\mathcal S}^*(M))$ that reduces both problems above to an analysis of the fibers of the spectral map ${\rm Spec}({\tt j})_2:{\rm Spec}({\mathcal S}^*(N))\to{\rm Spec}({\mathcal S}^*(M))$. If we denote $Z:={\rm cl}_{{\rm Spec}({\mathcal S}^*(M))}(M\setminus N)$, it holds that the restriction map ${\rm Spec}({\tt j})_2|:{\rm Spec}({\mathcal S}^*(N))\setminus{\rm Spec}({\tt j})_2^{-1}(Z)\to{\rm Spec}({\mathcal S}^*(M))\setminus Z$ is a homeomorphism. Our problem concentrates on the computation of the size of the fibers of ${\rm Spec}({\tt j})_2$ at the points of $Z$. The size of the fibers of prime ideals `close' to the complement $Y:=M\setminus N$ provides valuable information concerning how $N$ is immersed inside $M$. If $N$ is dense in $M$, the map ${\rm Spec}({\tt j})_2$ is surjective and the generic fiber of a prime ideal ${\mathfrak p}\in Z$ contains infinitely many elements. However, finite fibers may also appear and we provide a criterium to decide when the fiber ${\rm Spec}({\tt j})_2^{-1}({\mathfrak p})$ is a finite set for ${\mathfrak p}\in Z$.Comment: 33 pages, 3 figure

Gravity-driven instability in a spherical Hele-Shaw cell

A pair of concentric spheres separated by a small gap form a spherical Hele-Shaw cell. In this cell an interfacial instability arises when two immiscible fluids flow. We derive the equation of motion for the interface perturbation amplitudes, including both pressure and gravity drivings, using a mode coupling approach. Linear stability analysis shows that mode growth rates depend upon interface perimeter and gravitational force. Mode coupling analysis reveals the formation of fingering structures presenting a tendency toward finger tip-sharpening.Comment: 13 pages, 4 ps figures, RevTex, to appear in Physical Review

Social ties and economic development

We develop a parsimonious general equilibrium model where agents allocate time across three activities: production, trade, and leisure. Leisure includes time spent socializing, which economizes transaction costs. Our framework yields multiple equilibria in terms of the number of social ties and predicts that the number of social ties is positively associated with development, a relationship we observe in cross-country data. The model captures additional dimensions of data, namely: (i) increasing income inequality, but converging growth rates; (ii) an association between weak social ties and development; and (iii) an association between number of social ties and size of the transaction sector.social capital; development; transaction costs; networks

Transient glare: its effect on the lower threshold of motion

We measured the lower threshold of motion (LTM) of suprathreshold gratings as a function of spatial frequency and contrast, for both transient glare and no-glare conditions. A two alternatives forced choice paradigm, using the method of constant stimuli, was adopted to measure the LTM. The LTM occurs at constant velocity. This velocity threshold is higher for transient glare condition than for no-glare condition. We found that the sudden onset of glare increases LTM over the whole range of contrasts. We believe the effect of transient glare sources on the lower threshold of motion is due to the transient loss of sensitivity.Fil: Barraza, Jose Fernando. Universidad Nacional de Tucumán. Facultad de Ciencias Exactas y Tecnología. Departamento de Luminotecnia, Luz y Visión; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Tucumán. Instituto de Investigación en Luz, Ambiente y Visión. Universidad Nacional de Tucumán. Facultad de Ciencias Exactas y Tecnología. Instituto de Investigación en Luz, Ambiente y Visión; ArgentinaFil: Colombo, Elisa Margarita. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Tucumán. Instituto de Investigación en Luz, Ambiente y Visión. Universidad Nacional de Tucumán. Facultad de Ciencias Exactas y Tecnología. Instituto de Investigación en Luz, Ambiente y Visión; Argentina. Universidad Nacional de Tucumán. Facultad de Ciencias Exactas y Tecnología. Departamento de Luminotecnia, Luz y Visión; Argentin