Geometry of quasi-sum production functions with constant elasticity of substitution property

A production function $f$ is called quasi-sum if there are strict monotone functions $F, h_1,...,h_n$ with $F'>0$ such that $$f(x)= F(h_1 (x_1)+...+h_n (x_n)).$$ The justification for studying quasi-sum production functions is that these functions appear as solutions of the general bisymmetry equation and they are related to the problem of consistent aggregation. In this article, first we present the classification of quasi-sum production functions satisfying the constant elasticity of substitution property. Then we prove that if a quasi-sum production function satisfies the constant elasticity of substitution property, then its graph has vanishing Gauss-Kronecker curvature (or its graph is a flat space) if and only if the production function is either a linearly homogeneous generalized ACMS function or a linearly homogeneous generalized Cobb-Douglas function.Comment: 10 pages. Appeared in J. Adv. Math. Stud. 5 (2012), no. 2, 90-9

Open problems and conjectures on submanifolds of finite type revisited

Submanifolds of finite type were introduced by the author during the late 1970s. The first results on this subject had been collected in author's book [Total mean curvature and sub manifolds of finite type, World Scientific, NJ, 1984]. A list of ten open problems and three conjectures on submanifolds of finite type was published in 1981. The main purpose of this article is to provide some updated information on the three conjectures.Comment: 28 page

The Top Quark Forward Backward Asymmetry at CDF

It has been more than 15 years since the discovery of the top quark. Great strides have been made in the measurement of the top quark mass and the properties of it. Most results show consistency with the standard model. However, using 5 fb$^{-1}$ data, recent measurements of the asymmetry in the production of top and anti-top quark pair have demonstrated surprisingly large values at CDF. Using 4 fb$^{-1}$ data, D0 also has similar effect.Comment: 5 pages; for DIS 2011 conferenc

Statistical Inference with Local Optima

We study the statistical properties of an estimator derived by applying a gradient ascent method with multiple initializations to a multi-modal likelihood function. We derive the population quantity that is the target of this estimator and study the properties of confidence intervals (CIs) constructed from asymptotic normality and the bootstrap approach. In particular, we analyze the coverage deficiency due to finite number of random initializations. We also investigate the CIs by inverting the likelihood ratio test, the score test, and the Wald test, and we show that the resulting CIs may be very different. We provide a summary of the uncertainties that we need to consider while making inference about the population. Note that we do not provide a solution to the problem of multiple local maxima; instead, our goal is to investigate the effect from local maxima on the behavior of our estimator. In addition, we analyze the performance of the EM algorithm under random initializations and derive the coverage of a CI with a finite number of initializations. Finally, we extend our analysis to a nonparametric mode hunting problem.Comment: 66 page, 5 figure

Modal Regression using Kernel Density Estimation: a Review

We review recent advances in modal regression studies using kernel density estimation. Modal regression is an alternative approach for investigating relationship between a response variable and its covariates. Specifically, modal regression summarizes the interactions between the response variable and covariates using the conditional mode or local modes. We first describe the underlying model of modal regression and its estimators based on kernel density estimation. We then review the asymptotic properties of the estimators and strategies for choosing the smoothing bandwidth. We also discuss useful algorithms and similar alternative approaches for modal regression, and propose future direction in this field.Comment: 29 pages, 2 figures; a short invited review paper; new section on softwares for modal regressio

A tour through δ\delta-invariants: From Nash's embedding theorem to ideal immersions, best ways of living and beyond

First I will explain my motivation to introduce the $\delta$-invariants for Riemannian manifolds. I will also recall the notions of ideal immersions and best ways of living. Then I will present a few of the many applications of $\delta$-invariants to several areas in mathematics. Finally, I will present two optimal inequalities involving $\delta$-invariants for Lagrangian submanifolds obtained very recently in joint papers with F. Dillen, J. Van der Veken and L. Vrancken.Comment: 14 pages, to appear in a special volume of Publications de l'Institut Mathematique, Proceeding of XVII Geometrical Seminar, September 3-8, 2012, Zlatibor, Serbi

A Tutorial on Kernel Density Estimation and Recent Advances

This tutorial provides a gentle introduction to kernel density estimation (KDE) and recent advances regarding confidence bands and geometric/topological features. We begin with a discussion of basic properties of KDE: the convergence rate under various metrics, density derivative estimation, and bandwidth selection. Then, we introduce common approaches to the construction of confidence intervals/bands, and we discuss how to handle bias. Next, we talk about recent advances in the inference of geometric and topological features of a density function using KDE. Finally, we illustrate how one can use KDE to estimate a cumulative distribution function and a receiver operating characteristic curve. We provide R implementations related to this tutorial at the end.Comment: A tutorial paper; accepted to Biostatistics & Epidemiology. Main article: 26 pages, 8 figures. R implementations: 11 pages, generated by Rmarkdow

The 2-ranks of connected compact Lie groups

The 2-rank of a compact Lie group $G$ is the maximal possible rank of the elementary 2-subgroup ${\mathbb Z}_{2}\times... {\mathbb Z}_{2}$ of $G$. The study of 2-ranks (and $p$-rank for any prime $p$) of compact Lie groups was initiated in 1953 by A. Borel and J.-P. Serre. Since then the 2-ranks of compact Lie groups have been investigated by many mathematician. The 2-ranks of compact Lie groups relate closely with several important areas in mathematics. In this article, we survey important results concerning 2-ranks of compact Lie groups. In particular, we present the complete determination of 2-ranks of compact connected simple Lie groups $G$ via the 2-numbers introduced by B. Y. Chen and T. Nagano in [Un invariant g\'em\'etrique riemannien, C. R. Acad. Sci. Paris, 295 (1982), 389--391] and [A Riemannian geometric invariant and its applications to a problem of Borel and Serre, Trans. Amer. Math. Soc. 308 (1988), 273--297].Comment: 17 page

Covering maps and ideal embeddings of compact homogeneous spaces

The notion of ideal embeddings was introduced in [B.-Y. Chen, {Strings of Riemannian invariants, inequalities, ideal immersions and their applications.} The Third Pacific Rim Geometry Conference (Seoul, 1996), 7-60, Int. Press, Cambridge, MA, 1998]. Roughly speaking, an ideal embedding (or a best of living) is an isometrical embedding which receives the least possible amount of tension from the surrounding space at each point. In this article, we study ideal embeddings of irreducible compact homogenous spaces in Euclidean spaces via covering maps. Our main result states that $\pi: M\to N$ is a covering map between two irreducible compact homogeneous spaces and if $\lambda_1(M)\ne \lambda_1(N)$, then $N$ doesn't admit an ideal embedding in any Euclidean space; although $M$ could.Comment: 8 pages; to appear in Journal of Geometry and Symmetry in Physic

A survey on geometry of warped product submanifolds

The warped product $N_1\times_f N_2$ of two Riemannian manifolds $(N_1,g_1)$ and $(N_2,g_2)$ is the product manifold $N_1\times N_2$ equipped with the warped product metric $g=g_1+f^2 g_2$, where $f$ is a positive function on $N_1$. The notion of warped product manifolds is one of the most fruitful generalizations of Riemannian products. Such notion plays very important roles in differential geometry as well as in physics, especially in general relativity. Warped product manifolds have been studied for a long period of time. In contrast, the study of warped product submanifolds was only initiated around the beginning of this century. In this article we survey important results on warped product submanifolds in various ambient manifolds. It is the author's hope that this survey article will provide a good introduction on the theory of warped product submanifolds as well as a useful reference for further research on this vibrant research subject.Comment: 44 page