Data-Discriminants of Likelihood Equations

Maximum likelihood estimation (MLE) is a fundamental computational problem in statistics. The problem is to maximize the likelihood function with respect to given data on a statistical model. An algebraic approach to this problem is to solve a very structured parameterized polynomial system called likelihood equations. For general choices of data, the number of complex solutions to the likelihood equations is finite and called the ML-degree of the model. The only solutions to the likelihood equations that are statistically meaningful are the real/positive solutions. However, the number of real/positive solutions is not characterized by the ML-degree. We use discriminants to classify data according to the number of real/positive solutions of the likelihood equations. We call these discriminants data-discriminants (DD). We develop a probabilistic algorithm for computing DDs. Experimental results show that, for the benchmarks we have tried, the probabilistic algorithm is more efficient than the standard elimination algorithm. Based on the computational results, we discuss the real root classification problem for the 3 by 3 symmetric matrix~model.Comment: 2 table

Logarithmic concavity of Schur and related polynomials

We show that normalized Schur polynomials are strongly log-concave. As a consequence, we obtain Okounkov's log-concavity conjecture for Littlewood-Richardson coefficients in the special case of Kostka numbers

On the Statistical Distribution of Epidermal Papillomata in Mice

IN a previous investigation reported from this laboratory (Ball and McCarter, 1960) it was noted that tumours produced in the skin of the CFW mouse by treatment with 7,12-dimethylbenz(a)anthracene (DMBA) and croton oil, were not distributed among the mice in accordance with the expected Poisson's distribution. Animals bearing no tumours and those bearing many were much more numerous than expected. A quantitative analysis of induced primary adenomatous pulmonary tumours in mice was reported by Polissar and Shimkin (1954). They showed that the occurrence of such tumours was subject to Poisson's distribution and that deviations from this distribution could be attributed to heterogeneity of susceptibility in the animals. We have now analyzed the data obtained in our laboratory in three populations of mice undergoing epidermal carcinogenesis. MATERIALS AND METHODS Strain CFW.-These mice were females, purchased from Carworth Farms Inc., New City, New York. They were housed in groups of 10 in acrylic plastic boxes with stainless steel tops. The bedding was sawdust. Water and Purina Fox Chow Cubes were freely available. Strain CFW/D.-This strain originated when, through error, a male was included among the female CFW mice purchased from the supplier in 1958. Since that time, brother-sister mating has been carried out with a view to obtaining a single inbred line. Litters selected for brother-sister mating were chosen on the basis of health, number in the litter and even distribution of the sexes and not for sensitivity to carcinogenesis. The mice were in the thirteenth and fourteenth inbred generations when used. They were cared for as described above. Strain I.-This strain was obtained several years ago through the kindness of Dr. H. B. Andervont. The mice had been mated brother-to-sister for 71 to 72 generations when the experiment was begun. Chemicals. 7,1 2-dimethylbenz(a)anthracene and benzo((a)pyrene were obtained from Eastman Organic Chemicals. Croton oil was obtained from Bush and Co., Canada. Paraffin oil viscosity 125/135 NF was a product of Fisher Scientific Co., Montreal, Canada. Meprobamate (Miltown) was kindly supplied by Dr. F. M

Inverse Medea as a Novel Gene Drive System for Local Population Replacement: A Theoretical Analysis

One strategy to control mosquito-borne diseases, such as malaria and dengue fever, on a regional scale is to use gene drive systems to spread disease-refractory genes into wild mosquito populations. The development of a synthetic Medea element that has been shown to drive population replacement in laboratory Drosophila populations has provided encouragement for this strategy but has also been greeted with caution over the concern that transgenes may spread into countries without their consent. Here, we propose a novel gene drive system, inverse Medea, which is strong enough to bring about local population replacement but is unable to establish itself beyond an isolated release site. The system consists of 2 genetic components—a zygotic toxin and maternal antidote—which render heterozygous offspring of wild-type mothers unviable. Through population genetic analysis, we show that inverse Medea will only spread when it represents a majority of the alleles in a population. The element is best located on an autosome and will spread to fixation provided any associated fitness costs are dominant and to very high frequency otherwise. We suggest molecular tools that could be used to build the inverse Medea system and discuss its utility for a confined release of transgenic mosquitoes

Emerging sensor-cloud technology for pervasive services and applications

This is an Editorial article for the Special Issue on "Emerging Sensor-Cloud Technology for Pervasive Services and Applications" of the International Journal of Distributed Sensor Networks

Complements of hypersurfaces, variation maps and minimal models of arrangements

We prove the minimality of the CW-complex structure for complements of hyperplane arrangements in $\mathbb C^n$ by using the theory of Lefschetz pencils and results on the variation maps within a pencil of hyperplanes. This also provides a method to compute the Betti numbers of complements of arrangements via global polar invariants

Exact sub-grid interface correction schemes for elliptic interface problems

We introduce a non-conforming finite element method for second order elliptic interface problems. Our approach applies to problems in which discontinuous coefficients and singular sources on the interface may give rise to jump discontinuities in either the solution or its normal derivative. Given a standard background mesh and an interface that passes between elements, the key idea is to construct a singular correction function which satisfies the prescribed jump conditions, providing accurate sub-grid resolution of the discontinuities. Utilizing the closest point extension and an implicit interface representation by the signed distance function, an algorithm is established to construct the correction function. The result is a function which is supported only on the interface elements, represented by the regular basis functions, and bounded independently of the interface location with respect to the background mesh. In the particular case of a constant second order coefficient, our regularization by singular function is straightforward, and the resulting left-hand-side is identical to that of a regular problem without introducing any instability. The influence of the regularization appears solely on the right-hand-side, which simplifies the implementation. In the more general case of discontinuous second order coefficients, a normalization is invoked which introduces a constraint equation on the interface. This results in a problem statement similar to that of a saddle-point problem. We employ two-level-iteration as the solution strategy, which exhibits aspects similar to those of iterative preconditioning strategies