Polar Varieties, Real Equation Solving and Data-Structures: The hypersurface case

In this paper we apply for the first time a new method for multivariate equation solving which was developed in \cite{gh1}, \cite{gh2}, \cite{gh3} for complex root determination to the {\em real} case. Our main result concerns the problem of finding at least one representative point for each connected component of a real compact and smooth hypersurface. The basic algorithm of \cite{gh1}, \cite{gh2}, \cite{gh3} yields a new method for symbolically solving zero-dimensional polynomial equation systems over the complex numbers. One feature of central importance of this algorithm is the use of a problem--adapted data type represented by the data structures arithmetic network and straight-line program (arithmetic circuit). The algorithm finds the complex solutions of any affine zero-dimensional equation system in non-uniform sequential time that is {\em polynomial} in the length of the input (given in straight--line program representation) and an adequately defined {\em geometric degree of the equation system}. Replacing the notion of geometric degree of the given polynomial equation system by a suitably defined {\em real (or complex) degree} of certain polar varieties associated to the input equation of the real hypersurface under consideration, we are able to find for each connected component of the hypersurface a representative point (this point will be given in a suitable encoding). The input equation is supposed to be given by a straight-line program and the (sequential time) complexity of the algorithm is polynomial in the input length and the degree of the polar varieties mentioned above.Comment: Late

Polar Varieties and Efficient Real Elimination

Let $S_0$ be a smooth and compact real variety given by a reduced regular sequence of polynomials $f_1, ..., f_p$. This paper is devoted to the algorithmic problem of finding {\em efficiently} a representative point for each connected component of $S_0$ . For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of $S_0$. This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations $f_1, >..., f_p$ and in a suitably introduced, intrinsic geometric parameter, called the {\em degree} of the real interpretation of the given equation system $f_1, >..., f_p$.Comment: 32 page

Polar Varieties and Efficient Real Equation Solving: The Hypersurface Case

The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo \cite{gihemorpar} can be applied to a case of real polynomial equation solving. Our main result concerns the problem of finding one representative point for each connected component of a real bounded smooth hypersurface. The algorithm in \cite{gihemorpar} yields a method for symbolically solving a zero-dimensional polynomial equation system in the affine (and toric) case. Its main feature is the use of adapted data structure: Arithmetical networks and straight-line programs. The algorithm solves any affine zero-dimensional equation system in non-uniform sequential time that is polynomial in the length of the input description and an adequately defined {\em affine degree} of the equation system. Replacing the affine degree of the equation system by a suitably defined {\em real degree} of certain polar varieties associated to the input equation, which describes the hypersurface under consideration, and using straight-line program codification of the input and intermediate results, we obtain a method for the problem introduced above that is polynomial in the input length and the real degree.Comment: Late

How Do Antitobacco Campaign Advertising and Smoking Status Affect Beliefs and Intentions? Some Similarities and Differences Between Adults and Adolescents

This article presents two studies that examine similarities and differences with respect to how adults and adolescents process and respond to information in an antitobacco ad campaign. Study 1 examines (1) the effects of antitobacco advertising campaign measures (e.g., campaign advertisement integration, perceived strength of ad-based messages, attitude toward the ad campaign) on four key adult antismoking beliefs and (2) the influence of these campaign evaluations and beliefs on smokers’ intentions to quit smoking. Hierarchical regression results show that antismoking ad campaign reactions explain substantial additional variance in beliefs about tobacco industry deceptiveness, smoking addictiveness, harmfulness of secondhand smoke, and restrictions on smoking at different public venues. The findings also show that the campaign variables as a whole are positively related to intentions to quit smoking, beyond the variance that is explained by demographics. In Study 2, the authors replicate and extend these findings for the campaign using similar measures and procedures for a sample of more than 900 adolescents. They draw comparisons between these adult and adolescent findings and offer some implications for potential corrective advertising for consumers’ beliefs about smoking that may be required of tobacco companies based on U.S. v. Philip Morris USA, Inc

Serendipitous discovery of a projected pair of QSOs separated by 4.5 arcsec on the sky

We present the serendipitous discovery of a projected pair of quasi-stellar objects (QSOs) with an angular separation of $\Delta\theta =4.50$ arcsec. The redshifts of the two QSOs are widely different: one, our programme target, is a QSO with a spectrum consistent with being a narrow line Seyfert 1 AGN at $z=2.05$. For this target we detect Lyman-$\alpha$, \ion{C}{4}, and \ion{C}{3]}. The other QSO, which by chance was included on the spectroscopic slit, is a Type 1 QSO at a redshift of $z=1.68$, for which we detect \ion{C}{4}, \ion{C}{3]} and \ion{Mg}{2}. We compare this system to previously detected projected QSO pairs and find that only about a dozen previously known pairs have smaller angular separation.Comment: 4 pages, 3 figures. Accepted for publication in A

Extinction of cue-evoked food seeking recruits a GABAergic interneuron ensemble in the dorsal medial prefrontal cortex of mice

Animals must quickly adapt food-seeking strategies to locate nutrient sources in dynamically changing environments. Learned associations between food and environmental cues that predict its availability promote food-seeking behaviors. However, when such cues cease to predict food availability, animals undergo 'extinction' learning, resulting in the inhibition of food-seeking responses. Repeatedly activated sets of neurons, or 'neuronal ensembles', in the dorsal medial prefrontal cortex (dmPFC) are recruited following appetitive conditioning and undergo physiological adaptations thought to encode cue-reward associations. However, little is known about how the recruitment and intrinsic excitability of such dmPFC ensembles are modulated by extinction learning. Here, we used in vivo 2-Photon imaging in male Fos-GFP mice that express green fluorescent protein (GFP) in recently behaviorally-activated neurons to determine the recruitment of activated pyramidal and GABAergic interneuron mPFC ensembles during extinction. During extinction, we revealed a persistent activation of a subset of interneurons which emerged from a wider population of interneurons activated during the initial extinction session. This activation pattern was not observed in pyramidal cells, and extinction learning did not modulate the excitability properties of activated neurons. Moreover, extinction learning reduced the likelihood of reactivation of pyramidal cells activated during the initial extinction session. Our findings illuminate novel neuronal activation patterns in the dmPFC underlying extinction of food-seeking, and in particular, highlight an important role for interneuron ensembles in this inhibitory form of learning