Approximate Set Union Via Approximate Randomization

We develop an randomized approximation algorithm for the size of set union problem \arrowvert A_1\cup A_2\cup...\cup A_m\arrowvert, which given a list of sets $A_1,...,A_m$ with approximate set size $m_i$ for $A_i$ with $m_i\in \left((1-\beta_L)|A_i|, (1+\beta_R)|A_i|\right)$, and biased random generators with Prob(x=\randomElm(A_i))\in \left[{1-\alpha_L\over |A_i|},{1+\alpha_R\over |A_i|}\right] for each input set $A_i$ and element $x\in A_i,$ where $i=1, 2, ..., m$. The approximation ratio for \arrowvert A_1\cup A_2\cup...\cup A_m\arrowvert is in the range $[(1-\epsilon)(1-\alpha_L)(1-\beta_L), (1+\epsilon)(1+\alpha_R)(1+\beta_R)]$ for any $\epsilon\in (0,1)$, where $\alpha_L, \alpha_R, \beta_L,\beta_R\in (0,1)$. The complexity of the algorithm is measured by both time complexity, and round complexity. The algorithm is allowed to make multiple membership queries and get random elements from the input sets in one round. Our algorithm makes adaptive accesses to input sets with multiple rounds. Our algorithm gives an approximation scheme with O(\setCount\cdot(\log \setCount)^{O(1)}) running time and $O(\log m)$ rounds, where $m$ is the number of sets. Our algorithm can handle input sets that can generate random elements with bias, and its approximation ratio depends on the bias. Our algorithm gives a flexible tradeoff with time complexity O\left(\setCount^{1+\xi}\right) and round complexity $O\left({1\over \xi}\right)$ for any $\xi\in(0,1)$

An Urban Accounting for Geographic Concentration of Skills and Welfare Inequality

Using Jones (2014) generalized human capital accounting, we extend the urban accounting model of Desmet and Rossi-Hansberg (2013) to account for the geographic distribution of skills across US metropolitan areas. The methodology allows the productivity of high-skill workers to depend on location advantage and local skill mix; the latter also determines the productivity of low-skill workers. Urban friction, rising with population size, reduces worker consumption relative to their wage income. Amenities for high-skill and low-skill workers in each city are calibrated so that the utility for each skill type is equalized across cities. We examine counterfactual skill-mix distribution across cities and welfare gap between the skill groups by shutting down spatial heterogeneity in location advantage, amenity and excess friction respectively. We show that skill mix becomes more even across cities absent heterogeneity in location advantages or in excess friction but it becomes more dispersed absent amenity heterogeneity. The welfare gap widens when heterogeneity in any of the three factors is eliminated. The generalized urban accounting model can shed light on the causes of increased concentration of skilled workers in large cities in US highlighted by E. Moretti (2008) and Diamond (2012) and the implications for welfare gap between the skill groups

Land Use Rights, Government Land Supply, and the Pattern of Redevelopment in Shanghai

This paper reviews the urban redevelopment activities in Shanghai as the land market reforms were introduced. We focus on the impact of land use institutions on the spatial pattern of these activities. Sites for private real estate redevelopment were supplied by individual districts in the city. But the need for districts to pay for the resettlement of displaced residents contributed to a spatial mismatch between the supply of redevelopment sites and the market demand for commercial real estate space. Resettlement costs are highest at the high demand locations. State owned enterprises and institutions occupying land allocated by the state also engaged in real estate development. Whereas the density of private redevelopment was sensitive to the volume of commercial activities in a district, this does not appear to have been important in determining the location of the significant increase in the stock of commercial space resulting from development by local enterprises and institutions. This growth shows considerable decentralization between 1993 and 1996,indicative of spatially inefficient redevelopment activities by land-rich state enterprises.Institutions, Land Market, Economic Transition, Urban Redevelopment, Shanghai

Mating Siegel and Thurston quadratic polynomials

We prove that a quadratic polynomial with a bounded type Siegel disk and a quadratic post-critically finite polynomial are always mateable.Comment: 20 pages, 5 figure

Mating Siegel and parabolic quadratic polynomials

Let $f_\theta(z)=e^{2\pi i\theta}z+z^2$ be the quadratic polynomial having an indifferent fixed point at the origin. For any bounded type irrational number $\theta\in\mathbb{R}\setminus\mathbb{Q}$ and any rational number $\nu\in\mathbb{Q}$, we prove that $f_\theta$ and $f_\nu$ are conformally mateable, and that the mating is unique up to conjugacy by a M\"{o}bius map. This gives an affirmative (partial) answer to a question raised by Milnor in 2004. A crucial ingredient in the proof relies on an expansive property when iterating certain rational maps near Siegel disk boundaries. Combining this with the expanding property in repelling petals of parabolic points, we also prove that the Julia sets of a class of Siegel rational maps with parabolic points are locally connected.Comment: 40 pages, 9 figure

Do objects in working memory compete with objects in perception?

It is generally assumed that "perceptual object" is the basic unit for processing visual information and that only a small number of objects can be either perceptually selected or encoded in working memory (WM) at one time. This raises the question whether the same resource is used when objects are selected and tracked as when they are held in WM. In two experiments, we measured dual-task interference between a memory task and a Multiple Object Tracking task. The WM tasks involve explicit, implicit, or no spatial processing. Our results suggest there is no resource competition between working memory and perceptual selection except when the WM task requires encoding spatial properties