On vanishing sums of m \,m\,th roots of unity in finite fields

In an earlier work, the authors have determined all possible weights $n$ for which there exists a vanishing sum $\zeta_1+\cdots +\zeta_n=0$ of $m$th roots of unity $\zeta_i$ in characteristic 0. In this paper, the same problem is studied in finite fields of characteristic $p$. For given $m$ and $p$, results are obtained on integers $n_0$ such that all integers $n\geq n_0$ are in the ``weight set'' $W_p(m)$. The main result $(1.3)$ in this paper guarantees, under suitable conditions, the existence of solutions of $x_1^d+\cdots+x_n^d=0$ with all coordinates not equal to zero over a finite field

A Semi-parametric Two-component “Compound” Mixture Model and Its Application to Estimating Malaria Attributable Fractions

Malaria remains a major epidemiological problem in many developing countries. Malaria is dened as the presence of parasites and symptoms (usually fever) due to the parasites. In endemic areas, an individual may have symptoms attributable either to malaria or to other causes. From a clinical point of view, it is important to correctly diagnose an individual who has developed symptoms so that the appropriate treatments can be given. From an epidemiologic and economic point of view, it is important to determine the proportion of malaria affected cases in individuals who have symptoms so that policies on intervention programmes can be developed. Once symptoms have developed in an individual, the diagnosis of malaria can be based on analysis of the parasite levels in blood samples. However, even a blood test is not conclusive as in endemic areas, many healthy individuals can have parasites in their blood slides. Therefore, data from this type of studies can be viewed as coming from a mixture distribution, with the components corresponding to malaria and nonmalaria cases. A unique feature in this type of data, however, is the fact that a proportion of the non-malaria cases have zero parasite levels. Therefore, one of the component distribu-tions is itself a mixture distribution. In this article, we propose a semi-parametric likelihood approach for estimating the proportion of clinical malaria using parasite level data from a group of individuals with symptoms. Our approach assumes the density ratio for the parasite levels in clinical malaria and non-clinical malaria cases can be modeled using a logistic model. We use empirical likelihood to combine the zero and non-zero data. The maximum semi-parametric likelihood estimate is more ecient than existing non-parametric estimates using only the frequencies of zero and non-zero data. On the other hand, it is more robust than a fully parametric maximum likelihood estimate that assumes a parametric model for the non-zero data. Simulation results show that the performance of the proposed method is satisfactory. The proposed method is used to analyze data from a malaria survey carried out in Tanzania.Attributable fraction; Density ratio model; Empirical likelihood; Malaria; Mixture methods.

Comparing two financial crises: the case of Hong Kong real estate markets

Hong Kong is no stranger to bubbles or crisis. During the Asian Financial Crisis(AFC), the Hong Kong housing price index drops more than 50% in less than a year. The same market then experiences the Internet Bubble, the SARS attack, and recently the Global Financial Crisis (GFC). This paper attempts to provide some “stylized facts” of the real estate markets and the macroeconomy, and follow the event-study methodology to examine whether the markets behave differently in the AFC and GFC, and discuss the possible linkage to the change in government policies (“learning effect”) and the flow of Chinese consumers and investors to Hong Kong (“China factor”).regime switching, structural change, small open economy, bounded rationality, banking policy

Quasinormal Modes of Dirty Black Holes

Quasinormal mode (QNM) gravitational radiation from black holes is expected to be observed in a few years. A perturbative formula is derived for the shifts in both the real and the imaginary part of the QNM frequencies away from those of an idealized isolated black hole. The formulation provides a tool for understanding how the astrophysical environment surrounding a black hole, e.g., a massive accretion disk, affects the QNM spectrum of gravitational waves. We show, in a simple model, that the perturbed QNM spectrum can have interesting features.Comment: 4 pages. Published in PR

Financial Crisis and the Comovements of Housing Sub-markets: Do relationships change after a crisis?

This study of the co-movements of the transaction prices and trading volumes reveal that the mean correlation of prices, and trading volumes alike, among different housing sub-markets increases during the market boom. After a financial crisis, the correlations drop dramatically and stay low. The distribution of the correlations changes from skewed to symmetric. All these coincide with the increase in the total variance of prices, as well as the share of the idiosyncratic component in the total variance after the crisis. These findings are consistent to a family of theories which emphasize on “regime switch” in expectation.financial crisis; hedonic pricing; structural break; evolution of valuation; rolling regression