Application of ultra-performance liquid chromatography/tandem mass spectrometry for the measurement of vitamin D in foods and nutritional supplements

Ultra-performance liquid chromatography (UPLC)/MS/MS was applied to measure vitamin D in various foods and nutritional supplements. The run-time of the chromatographic separation was cut from 20 min in HPLC/MS/MS to 10 min in UPLC/MS/MS, while equal or better separation efficiency was achieved to deal with complex food matrixes. Under the optimized conditions, all the previtamins of vitamin D3, D2, and isotope-labeled vitamin D3 were baseline-separated from their corresponding vitamins. It was also demonstrated that many sterol isomers in complex food matrixes that interfere in the analysis could be well-separated from the analytes. Accuracy of this method was evaluated by analysis of NIST SRM 1849 infant formula reference material. With eight replicates, the average vitamin D3 concentration was 0.251 +/- 0.012 mg/kg, an excellent agreement with the certified value of 0.251 +/- 0.027 mg/kg. In addition, spike recovery from a commercial infant formula matrix was in the range of 100 to 108% for both vitamins D3 and D2 at three spike concentration levels. The spike recovery for an even more complex matrix, pet food, was 101-105%. LOQ values were 0.026 and 0.033 IU/g, or 0.086 and 0.11 IU/mL in solution, for vitamins D3 and D2, respectively. The dynamic range had three orders of magnitude, which made the method flexible and useful to deal with the wide concentration range of vitamin D in various samples. The method was robust based on the results of changing the parameters of LC separation and MS measurement. This accurate and reliable vitamin D method increased instrument efficiency and analysis productivity significantly.     

Generalizations of Polya's urn Problem

We consider generalizations of the classical Polya urn problem: Given finitely many bins each containing one ball, suppose that additional balls arrive one at a time. For each new ball, with probability p, create a new bin and place the ball in that bin; with probability 1–p, place the ball in an existing bin, such that the probability that the ball is placed in a bin is proportional to $ m^\gamma $, where m is the number of balls in that bin. For p=0, the number of bins is fixed and finite, and the behavior of the process depends on whether is greater than, equal to, or less than 1. We survey the known results and give new proofs for all three cases. We then consider the case p>0. When =1, this is equivalent to the so-called preferential attachment scheme which leads to power law distribution for bin sizes. When >1, we prove that a single bin dominates, i.e., as the number of balls goes to infinity, the probability that any new ball either goes into that bin or creates a new bin converges to 1. When p > 0 and < 1, we show that under the assumption that certain limits exist, the fraction of bins having m balls shrinks exponentially as a function of m. We then discuss further generalizations and pose several open problems.AMS Subject Classification: 05D40, 60C05, 60G20, 68R10, 91C99.     

Homologous non-isotopic symplectic surfaces of higher genus

We construct an infinite family of homologous, non-isotopic, symplectic surfaces of any genus greater than one in a certain class of closed, simply connected, symplectic four-manifolds. Our construction is the first example of this phenomenon for surfaces of genus greater than one.

     

Lagrangian tori in homotopy elliptic surfaces

Let denote the symplectic four-manifold, homotopy equivalent to the rational elliptic surface, corresponding to a fibred knot in constructed by R. Fintushel and R. J. Stern in 1998. We construct a family of nullhomologous Lagrangian tori in and prove that infinitely many of these tori have complements with mutually non-isomorphic fundamental groups if the Alexander polynomial of has some irreducible factor which does not divide for any positive integer . We also show how these tori can be non-isotopically embedded as nullhomologous Lagrangian submanifolds in other symplectic -manifolds.

     

The Kauffman bracket skein as an algebra of observables

We prove that the Kauffman bracket skein algebra of a cylinder over a surface with boundary, defined over complex numbers, is isomorphic to the observables of an appropriate lattice gauge field theory.

     

Uncertainty Quantification for Modern High-Dimensional Regression via Scalable Bayesian Methods

Tremendous progress has been made in the last two decades in the area of high-dimensional regression, especially in the “large p, small n” setting. Such sample starved settings inevitably lead to models which are potentially very unstable and hence quite unreliable. To this end, Bayesian shrinkage methods have generated a lot of recent interest in the modern high-dimensional regression and model selection context. Such methods span the wide spectrum of modern regression approaches and include among others, spike-and-slab priors, the Bayesian lasso, ridge regression, and global-local shrinkage priors such as the Horseshoe prior and the Dirichlet–Laplace prior. These methods naturally facilitate tractable uncertainty quantification and have thus been used extensively across diverse applications. A common unifying feature of these models is that the corresponding priors on the regression coefficients can be expressed as a scale mixture of normals. This property has been leveraged extensively to develop various three-step Gibbs samplers to explore the corresponding intractable posteriors. The convergence of such samplers however is very slow in high dimensions settings, making them disconnected to the very setting that they are intended to work in. To address this challenge, we propose a comprehensive and unifying framework to draw from the same family of posteriors via a class of tractable and scalable two-step blocked Gibbs samplers. We demonstrate that our proposed class of two-step blocked samplers exhibits vastly superior convergence behavior compared to the original three-step sampler in high-dimensional regimes on simulated data as well as data from a variety of applications including gene expression data, infrared spectroscopy data, and socio-economic/law enforcement data. We also provide a detailed theoretical underpinning to the new method by deriving explicit upper bounds for the (geometric) rate of convergence, and by proving that the proposed two-step sampler has superior spectral properties. Supplementary material for this article is available online.