Chromatic Zeros On Hierarchical Lattices and Equidistribution on Parameter Space

Associated to any finite simple graph $\Gamma$ is the chromatic polynomial $P_\Gamma(q)$ whose complex zeroes are called the chromatic zeros of $\Gamma$. A hierarchical lattice is a sequence of finite simple graphs $\{\Gamma_n\}_{n=0}^\infty$ built recursively using a substitution rule expressed in terms of a generating graph. For each $n$, let $\mu_n$ denote the probability measure that assigns a Dirac measure to each chromatic zero of $\Gamma_n$. Under a mild hypothesis on the generating graph, we prove that the sequence $\mu_n$ converges to some measure $\mu$ as $n$ tends to infinity. We call $\mu$ the limiting measure of chromatic zeros associated to $\{\Gamma_n\}_{n=0}^\infty$. In the case of the Diamond Hierarchical Lattice we prove that the support of $\mu$ has Hausdorff dimension two. The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications.Comment: To appear in Annales de l'Institut Henri Poincar\'e D. We have added considerably more background on activity currents and especially on the Dujardin-Favre classification of the passive locus. Exposition in the proof of the main theorem was improved. Comments welcome

The Value of Failures in Pharmaceutical R&D

We build a cumulative innovation model in which both success and failure provide valuable information for future research. To test this learning mechanism, we use a dataset covering outcomes of world-wide R&D projects in the pharmaceutical industry, and proxy knowledge flows with forward citations received by patents associated with each project. Empirical results confirm theoretical predictions that patents associated with successfully completed projects (i.e., leading to drug launch on the market) receive more citations than those associated to failed (terminated) projects, which in turn are cited more often than patents lacking clinical or preclinical information. We therefore offer evidence of the value of failures as research inputs in (pharmaceutical) innovationR&D competition, patent policy, pharmaceutical industry

The Appearance of the Rural in China's Tourism

State-led programs for rural development through tourism serve to reaffirm and reinstate rural spaces as the ideal periphery, a desirable and attractive decorative edge. to the modern, contemporary Chinese nation. By tracing the ways in which tourism development both centralizes the necessity of modernizing rural regions for the nation as a whole while simultaneously emphasizing the otherness. of rural communities in order to promote them as tourist attractions, in this essay I seek to understand how the rural. appears in Chinese tourism as a vital concern of the state by characterizing what is rural as increasingly different and distant in order to satisfy perceived tourist desires. In particular, the Chinese state represents village-based tourism projects through discourses of distance that render the rural. both absolutely critical to national processes of development and fundamentally peripheral to modern conditions

Wide-band variable transversal phase-shifter

We present a novel broadband phase-shifter based on a transversal filter configuration. This approach allows flexible control of the amplitude response while providing continuous variation of a linear phase slope. Numerical examples, both ideal and using practical RF components are presented and practical challenges in realising the phase-shifter are identified

Wideband RF photonic vector sum phase-shifter

A novel broadband linear phase phase-shifter based on the vector summation method is proposed. A photonic implementation of the phase-shifter with a continuously variable linear phase-shift up to 120° over the frequency range of DC-4 GHz is demonstrated. Good agreement between the measured responses and theoretical predictions is obtained