Fitting Effective Diffusion Models to Data Associated with a "Glassy Potential": Estimation, Classical Inference Procedures and Some Heuristics

A variety of researchers have successfully obtained the parameters of low dimensional diffusion models using the data that comes out of atomistic simulations. This naturally raises a variety of questions about efficient estimation, goodness-of-fit tests, and confidence interval estimation. The first part of this article uses maximum likelihood estimation to obtain the parameters of a diffusion model from a scalar time series. I address numerical issues associated with attempting to realize asymptotic statistics results with moderate sample sizes in the presence of exact and approximated transition densities. Approximate transition densities are used because the analytic solution of a transition density associated with a parametric diffusion model is often unknown.I am primarily interested in how well the deterministic transition density expansions of Ait-Sahalia capture the curvature of the transition density in (idealized) situations that occur when one carries out simulations in the presence of a "glassy" interaction potential. Accurate approximation of the curvature of the transition density is desirable because it can be used to quantify the goodness-of-fit of the model and to calculate asymptotic confidence intervals of the estimated parameters. The second part of this paper contributes a heuristic estimation technique for approximating a nonlinear diffusion model. A "global" nonlinear model is obtained by taking a batch of time series and applying simple local models to portions of the data. I demonstrate the technique on a diffusion model with a known transition density and on data generated by the Stochastic Simulation Algorithm.Comment: 30 pages 10 figures Submitted to SIAM MMS (typos removed and slightly shortened

Approximate closed-form formulas for the zeros of the Bessel Polynomials

We find approximate expressions x(k,n) and y(k,n) for the real and imaginary parts of the kth zero z_k=x_k+i y_k of the Bessel polynomial y_n(x). To obtain these closed-form formulas we use the fact that the points of well-defined curves in the complex plane are limit points of the zeros of the normalized Bessel polynomials. Thus, these zeros are first computed numerically through an implementation of the electrostatic interpretation formulas and then, a fit to the real and imaginary parts as functions of k and n is obtained. It is shown that the resulting complex number x(k,n)+i y(k,n) is O(1/n^2)-convergent to z_k for fixed kComment: 9 pages, 2 figure

On the logarithmic comparison theorem for integrable logarithmic connections

Let $X$ be a complex analytic manifold, $D\subset X$ a free divisor with jacobian ideal of linear type (e.g. a locally quasi-homogeneous free divisor), $j: U=X-D \to X$ the corresponding open inclusion, $E$ an integrable logarithmic connection with respect to $D$ and $L$ the local system of the horizontal sections of $E$ on $U$. In this paper we prove that the canonical morphisms between the logarithmic de Rham complex of $E(kD)$ and $R j_* L$ (resp. the logarithmic de Rham complex of $E(-kD)$ and $j_!L$) are isomorphisms in the derived category of sheaves of complex vector spaces for $k\gg 0$ (locally on $X$)Comment: Terminology has changed: "linear jacobian type" instead of "commutative differential type"); no Koszul hypothesis is needed in theorem (2.1.1); minor changes. To appear in Proc. London Math. So

Survival and Nonescape Probabilities for Resonant and Nonresonant Decay

In this paper we study the time evolution of the decay process for a particle confined initially in a finite region of space, extending our analysis given recently (Phys. Rev. Lett. 74, 337 (1995)). For this purpose, we solve exactly the time-dependent Schroedinger equation for a finite-range potential. We calculate and compare two quantities: (i) the survival probability S(t), i.e., the probability that the particle is in the initial state after a time t; and (ii) the nonescape probability P(t), i.e., the probability that the particle remains confined inside the potential region after a time t. We analyze in detail the resonant and nonresonant decay. In the former case, after a very short time, S(t) and P(t) decay exponentially, but for very long times they decay as a power law, albeit with different exponents. For the nonresonant case we obtain that both quantities differ initially. However, independently of the resonant and nonresonant character of the initial state we always find a transition to the ground state of the system which indicates a process of ``loss of memory'' in the decay.Comment: 26 pages, RevTex file, figures available upon request from [email protected] (To be published in Annals of Physics

SPITZER observations of the λ Orionis cluster. II. Disks around solar-type and low-mass stars

We present IRAC/MIPS Spitzer Space Telescope observations of the solar-type and the low-mass stellar population of the young (~5Myr) λ Orionis cluster. Combining optical and Two Micron All Sky Survey photometry, we identify 436 stars as probable members of the cluster. Given the distance (450 pc) and the age of the cluster, our sample ranges in mass from 2 M_⊙ to objects below the substellar limit. With the addition of the Spitzer mid-infrared data, we have identified 49 stars bearing disks in the stellar cluster. Using spectral energy distribution slopes, we place objects in several classes: non-excess stars (diskless), stars with optically thick disks, stars with “evolved disks” (with smaller excesses than optically thick disk systems), and “transitional disk” candidates (in which the inner disk is partially or fully cleared). The disk fraction depends on the stellar mass, ranging from ~6% for K-type stars (R_C − J 4). We confirm the dependence of disk fraction on stellar mass in this age range found in other studies. Regarding clustering levels, the overall fraction of disks in the λ Orionis cluster is similar to those reported in other stellar groups with ages normally quoted as ~5Myr

Searching for star-forming dwarf galaxies in the Antlia cluster

The formation and evolution of dwarf galaxies in clusters need to be understood, and this requires large aperture telescopes. In this sense, we selected the Antlia cluster to continue our previous work in the Virgo, Fornax, and Hydra clusters and in the Local Volume (LV). Because of the scarce available literature data, we selected a small sample of five blue compact dwarf (BCD) candidates in Antlia for observation. Using the Gemini South and GMOS camera, we acquired the Halpha imaging needed to detect star-forming regions in this sample. With the long-slit spectroscopic data of the brightest seven knots detected in three BCD candidates, we derived their basic chemical properties. Using archival VISTA VHS survey images, we derived K_S magnitudes and surface brightness profile fits for the whole sample to assess basic physical properties. FS90-98, FS90-106, and FS90-147 are confirmed as BCDs and cluster members, based on their morphology, K_S surface photometry, oxygen abundance, and velocity redshift. FS90-155 and FS90-319 did not show any H{\alpha} emission, and they could not be confirmed as dwarf cluster star-forming galaxies. Based on our data, we studied some fundamental relations to compare star forming dwarfs (BCDs and dIs) in the LV and in the Virgo, Fornax, Hydra, and Antlia clusters. Star-forming dwarfs in nearby clusters appear to follow same fundamental relations in the near infrared with similar objects in the LV, specifically the size-luminosity and the metallicity-luminosity, while other more fundamental relations could not be checked in Antlia due to lack of data.Comment: Accepted for publication in A&A (early 2014