The moduli space of matroids

In the first part of the paper, we clarify the connections between several algebraic objects appearing in matroid theory: both partial fields and hyperfields are fuzzy rings, fuzzy rings are tracts, and these relations are compatible with the respective matroid theories. Moreover, fuzzy rings are ordered blueprints and lie in the intersection of tracts with ordered blueprints; we call the objects of this intersection pastures. In the second part, we construct moduli spaces for matroids over pastures. We show that, for any non-empty finite set $E$, the functor taking a pasture $F$ to the set of isomorphism classes of rank-$r$ $F$-matroids on $E$ is representable by an ordered blue scheme $Mat(r,E)$, the moduli space of rank-$r$ matroids on $E$. In the third part, we draw conclusions on matroid theory. A classical rank-$r$ matroid $M$ on $E$ corresponds to a $\mathbb{K}$-valued point of $Mat(r,E)$ where $\mathbb{K}$ is the Krasner hyperfield. Such a point defines a residue pasture $k_M$, which we call the universal pasture of $M$. We show that for every pasture $F$, morphisms $k_M\to F$ are canonically in bijection with $F$-matroid structures on $M$. An analogous weak universal pasture $k_M^w$ classifies weak $F$-matroid structures on $M$. The unit group of $k_M^w$ can be canonically identified with the Tutte group of $M$. We call the sub-pasture $k_M^f$ of $k_M^w$ generated by ``cross-ratios' the foundation of $M$,. It parametrizes rescaling classes of weak $F$-matroid structures on $M$, and its unit group is coincides with the inner Tutte group of $M$. We show that a matroid $M$ is regular if and only if its foundation is the regular partial field, and a non-regular matroid $M$ is binary if and only if its foundation is the field with two elements. This yields a new proof of the fact that a matroid is regular if and only if it is both binary and orientable.Comment: 83 page

Descartes' rule of signs, Newton polygons, and polynomials over hyperfields

We develop a theory of multiplicities of roots for polynomials over hyperfields and use this to provide a unified and conceptual proof of both Descartes' rule of signs and Newton's "polygon rule".Comment: 21 pages. v2: Revised the exposition and organization of the paper, corrected some minor typos. v3: Revised according to referee report, added new reference

On a theorem of Lafforgue

We give a new proof, along with some generalizations, of a folklore theorem (attributed to Laurent Lafforgue) that a rigid matroid (i.e., a matroid with indecomposable basis polytope) has only finitely many projective equivalence classes of representations over any given field.Comment: 11 page

Stress-Induced Cocaine Seeking Requires a Beta-2 Adrenergic Receptor-Regulated Pathway from the Ventral Bed Nucleus of the Stria Terminalis That Regulates CRF Actions in the Ventral Tegmental Area

The ventral bed nucleus of the stria terminalis (vBNST) has been implicated in stress-induced cocaine use. Here we demonstrate that, in the vBNST, corticotropin releasing factor (CRF) is expressed in neurons that innervate the ventral tegmental area (VTA), a site where the CRF receptor antagonist antalarmin prevents the reinstatement of cocaine seeking by a stressor, intermittent footshock, following intravenous self-administration in rats. The vBNST receives dense noradrenergic innervation and expresses β adrenergic receptors (ARs). Footshock-induced reinstatement was prevented by bilateral intra-vBNST injection of the β-2 AR antagonist, ICI-118,551, but not the β-1 AR antagonist, betaxolol. Moreover, bilateral intra-vBNST injection of the β-2 AR agonist, clenbuterol, but not the β-1 agonist, dobutamine, reinstated cocaine seeking, suggesting that activation of vBNST β-2 AR is both necessary for stress-induced reinstatement and sufficient to induce cocaine seeking. The contribution of a β-2 AR-regulated vBNST-to-VTA pathway that releases CRF was investigated using a disconnection approach. Injection of ICI-118,551 into the vBNST in one hemisphere and antalarmin into the VTA of the contralateral hemisphere prevented footshock-induced reinstatement, whereas ipsilateral manipulations failed to attenuate stress-induced cocaine seeking, suggesting that β-2 AR regulate vBNST efferents that release CRF into the VTA, activating CRF receptors, and promoting cocaine use. Last, reinstatement by clenbuterol delivered bilaterally into the vBNST was prevented by bilateral vBNST pretreatment with antalarmin, indicating that β-2 AR-mediated actions in the vBNST also require local CRF receptor activation. Understanding the processes through which stress induces cocaine seeking should guide the development of new treatments for addiction

Antagonism of GABA-B but not GABA-A receptors in the VTA prevents stress- and intra-VTA CRF-induced reinstatement of extinguished cocaine seeking in rats

Stress-induced reinstatement of cocaine seeking requires corticotropin releasing factor (CRF) actions in the ventral tegmental area (VTA). However the mechanisms through which CRF regulates VTA function to promote cocaine use are not fully understood. Here we examined the role of GABAergic neurotransmission in the VTA mediated by GABA-A or GABA-B receptors in the reinstatement of extinguished cocaine seeking by a stressor, uncontrollable intermittent footshock, or bilateral intra-VTA administration of CRF. Rats underwent repeated daily cocaine self-administration (1.0 mg/kg/ing; 14 × 6 h/day) and extinction and were tested for reinstatement in response to footshock (0.5 mA, 0.5” duration, average every 40 s; range 10–70 s) or intra-VTA CRF delivery (500 ng/side) following intra-VTA pretreatment with the GABA-A antagonist, bicuculline, the GABA-B antagonist, 2-hydroxysaclofen or vehicle. Intra-VTA bicuculline (1, 10 or 20 ng/side) failed to block footshock- or CRF-induced cocaine seeking at either dose tested. By contrast, 2-hydroxysaclofen (0.2 or 2 μg/side) prevented reinstatement by both footshock and intra-VTA CRF at a concentration that failed to attenuate food-reinforced lever pressing (45 mg sucrose-sweetened pellets; FR4 schedule) in a separate group of rats. These data suggest that GABA-B receptor-dependent CRF actions in the VTA mediate stress-induced cocaine seeking and that GABA-B receptor antagonists may have utility for the management of stress-induced relapse in cocaine addicts

Establishing Priorities for Sustainable Environmental Design in the Rural Villages of Yunnan, China

This paper addresses sustainable rural village development in China. Rural development is unlike the process of urbanization in Chinese cities and reflects different land ownership rules and different organizational structures. Even though there are an increasing number of Chinese residents in cities, there are still more than 600 million people living in the countryside. The attention lavished on city development has been, in part, now refocused to rural villages. Since 2006, the support for large-scale investment in the countryside has created much change; however, not all of this change is well organized, with potential for less than optimum impacts on the environment and sustainability. The paper identifies the key influences and drivers from historic and contemporary points of view. The sustainability of the villages will derive from long-term self-sufficiency, and this must include the understanding of environmental design principles, which enable suitable dwelling design. Two villages are taken as contrasting examples, and information derived from other sources is discussed. Technologies and techniques that can help determine environmental design priorities are evaluated and directions for future development suggested. This includes development of a design support aid with key drivers of: orientation and site location, window design and key construction features

Foundations of matroids -- Part 2: Further theory, examples, and computational methods

In this sequel to "Foundations of matroids - Part 1", we establish several presentations of the foundation of a matroid in terms of small building blocks. For example, we show that the foundation of a matroid M is the colimit of the foundations of all embedded minors of M isomorphic to one of the matroids $U^2_4$, $U^2_5$, $U^3_5$, $C_5$, $C_5^\ast$, $U^2_4\oplus U^1_2$, $F_7$, $F_7^\ast$, and we show that this list is minimal. We establish similar minimal lists of building blocks for the classes of 2-connected and 3-connected matroids. We also establish a presentation for the foundation of a matroid in terms of its lattice of flats. Each of these presentations provides a useful method to compute the foundation of certain matroids, as we illustrate with a number of concrete examples. Combining these techniques with other results in the literature, we are able to compute the foundations of several interesting classes of matroids, including whirls, rank-2 uniform matroids, and projective geometries. In an appendix, we catalogue various 'small' pastures which occur as foundations of matroids, most of which were found with the assistance of a computer, and we discuss some of their interesting properties.Comment: 69 page