Greg Huber

Bio: Greg Huber is an academic researcher from Massachusetts Institute of Technology. The author has contributed to research in topics: Gamma function. The author has an hindex of 2, co-authored 3 publications receiving 48 citations.

Gamma Function Derivation of n-Sphere Volumes

Abstract: (1982). Gamma Function Derivation of n-Sphere Volumes. The American Mathematical Monthly: Vol. 89, No. 5, pp. 301-302.

The n-Cube is Rupert

Abstract: An oval in is called Rupert if a straight tunnel can be made in it through which a second congruent oval can be passed. We show that the n-cube is Rupert for each n ⩾ 3.

Nonlocal Diffusion and Applications

Abstract: Introduction.- 1 A probabilistic motivation.-1.1 The random walk with arbitrarily long jumps.- 1.2 A payoff model.-2 An introduction to the fractional Laplacian.-2.1 Preliminary notions.- 2.2 Fractional Sobolev Inequality and Generalized Coarea Formula.- 2.3 Maximum Principle and Harnack Inequality.- 2.4 An s-harmonic function.- 2.5 All functions are locally s-harmonic up to a small error.- 2.6 A function with constant fractional Laplacian on the ball.- 3 Extension problems.- 3.1 Water wave model.- 3.2 Crystal dislocation.- 3.3 An approach to the extension problem via the Fourier transform.- 4 Nonlocal phase transitions.- 4.1 The fractional Allen-Cahn equation.- 4.2 A nonlocal version of a conjecture by De Giorgi.- 5 Nonlocal minimal surfaces.- 5.1 Graphs and s-minimal surfaces.- 5.2 Non-existence of singular cones in dimension 2 5.3 Boundary regularity.- 6 A nonlocal nonlinear stationary Schrodinger type equation.- 6.1 From the nonlocal Uncertainty Principle to a fractional weighted inequality.- Alternative proofs of some results.- A.1 Another proof of Theorem A.2 Another proof of Lemma 2.3.- References.

Confidence Analysis of Standard Deviational Ellipse and Its Extension into Higher Dimensional Euclidean Space

Abstract: Standard deviational ellipse (SDE) has long served as a versatile GIS tool for delineating the geographic distribution of concerned features. This paper firstly summarizes two existing models of calculating SDE, and then proposes a novel approach to constructing the same SDE based on spectral decomposition of the sample covariance, by which the SDE concept is naturally generalized into higher dimensional Euclidean space, named standard deviational hyper-ellipsoid (SDHE). Then, rigorous recursion formulas are derived for calculating the confidence levels of scaled SDHE with arbitrary magnification ratios in any dimensional space. Besides, an inexact-newton method based iterative algorithm is also proposed for solving the corresponding magnification ratio of a scaled SDHE when the confidence probability and space dimensionality are pre-specified. These results provide an efficient manner to supersede the traditional table lookup of tabulated chi-square distribution. Finally, synthetic data is employed to generate the 1-3 multiple SDEs and SDHEs. And exploratory analysis by means of SDEs and SDHEs are also conducted for measuring the spread concentrations of Hong Kong’s H1N1 in 2009.

An exploration of the equivalent weights particle filter

Abstract: Particle filters are fully nonlinear data assimilation techniques that aim to represent the probability distribution of the model state given the observations (the posterior) by a number of particles. In high-dimensional geophysical applications, the number of particles required by the sequential importance resampling (SIR) particle filter (in order to capture the high-probability region of the posterior) is too large to make them usable. However particle filters can be formulated using proposal densities, which give greater freedom in how particles are sampled and allow for a much smaller number of particles. Here a particle filter is presented which uses the proposal density to ensure that all particles end up in the high-probability region of the posterior probability density function. This gives rise to the possibility of nonlinear data assimilation in large-dimensional systems. The particle filter formulation is compared to the optimal proposal density particle filter and the implicit particle filter, both of which also utilise a proposal density. We show that, when observations are available every time step, both schemes will be degenerate when the number of independent observations is large, unlike the new scheme. The sensitivity of the new scheme to its parameter values is explored theoretically and demonstrated using the Lorenz (1963) model. Copyright © 2012 Royal Meteorological Society

Refinements, Generalizations, and Applications of Jordan's Inequality and Related Problems

Abstract: This is a survey and expository article. Some new developments on refinements, generalizations, and applications of Jordan's inequality and related problems, including some results about Wilker-Anglesio's inequality, some estimates for three kinds of complete elliptic integrals, and several inequalities for the remainder of power series expansion of the exponential function, are summarized.

Solving the Trust-Region Subproblem By a Generalized Eigenvalue Problem

Abstract: The state-of-the-art algorithms for solving the trust-region subproblem (TRS) are based on an iterative process, involving solutions of many linear systems, eigenvalue problems, subspace optimization, or line search steps. A relatively underappreciated fact, due to Gander, Golub, and von Matt [Linear Algebra Appl., 114 (1989), pp. 815--839], is that TRSs can be solved by one generalized eigenvalue problem, with no outer iterations. In this paper we rediscover this fact and discover its great practicality, which exhibits good performance both in accuracy and efficiency. Moreover, we generalize the approach in various directions, namely by allowing for an ellipsoidal constraint, dealing with the so-called hard case, and obtaining approximate solutions efficiently when high accuracy is unnecessary. We demonstrate that the resulting algorithm is a general-purpose TRS solver, effective both for dense and large-sparse problems, including the so-called hard case. Our algorithm is easy to implement: its essence i...