A result on hypothesis testing for a multivariate normal distribution when some observations are missing

Let U_i = (X_i, Y_i), i = 1, 2,…, n, be a random sample from a bivariate normal distribution with mean μ = (μ_x, μ_y) and covariance matrix

. Let X_i, i = n + 1,…, N represent additional independent observations on the X population. Consider the hypothesis testing problem H₀ : μ = 0 vs. H₁ : μ ≠ 0. We prove that Hotelling's T² test, which uses (X_i, Y_i), i = 1, 2,…, n (and discards X_i, i = n + 1,…, N) is an admissible test. In addition, and from a practical point of view, the proof will enable us to identify the region of the parameter space where the T²-test cannot be beaten. A similar result is also proved for the problem of testing μ_x ? μ_y = 0. A Bayes test and other competitors which are similar tests are discussed.     

Fluorierung von Cyanurchlorid und Tieftemperatur-Kristallstruktur von [(ClCN)3F]+ [AsF6]−

Fluorination of Cyanuric Chloride and Low-Temperature Crystal Structure of [(ClCN)₃F]⁺[AsF₆]^? The low-temperature fluorination of cyanuric chloride, (ClCN)₃, with F₂/AsF₅ in SO₂F₂ solution yielded the salt [(ClCN)₃F]⁺ [AsF₆]^? ( 1 ) essentially in quantitative yield. Compound 1 was identified by a low-temperature single crystal X-ray structure determination: R 3 c, trigonal, a = b = 10.4246(23) Å, c = 15.1850(24) Å, V = 1429.1(4) Å 3, Z = 6, R_F = 0.056, R_w = 0.076 (for significant reflections), R_F = 0.088, R_w = 0.079 (for all reflections). Fluorination of neat (ClCN)₃ with [NF₄]⁺ [Sb₂F₁₁]^? yielded NF₃, CClF₃, SbF₃, N₂ and traces of CF₄. A qualitative scale for the oxidizing strength of the oxidative fluorinators NF₄⁺ and (XCN)₃F⁺ (X = H, F, Cl) has been computed ab initio.     

Correlation between excess adsorption data measured for solvent mixture/adsorbent systems and RM values obtained for different solutes chromatographed in binary mobile phases

This paper concerns the application of excess adsorption isotherms, measured for solvent mixture/adsorbent systems, to the characterization of TLC data. For this purpose the excess adsorption isotherms for three liquid mixtures: cyclohexane/ benzene, benzene/acetone, and carbon tetrachloride/ethyl acetate on silica gel at 20°C have been measured. These mixtures have been used as binary mobile phases in TLC measurements. It has been shown for a given solute in binary mobile phase that the quantity R_M is a simple function of the excess adsorption. Parameters of this function have been used to characterize chromatographic systems with binary mobile phases.     

ADSORPTION OF 2,4-D ON MODIFIED HYPERCROSSLINKED POLYSTYRENE （NDA-99） AND XAD-4 RESIN

The adsorption behavior of pesticide 2,4-dichlorophenoxyacetic acid (2,4-D) in aqueous solution has been investigated using a hypercrosslinked polystyrene adsorbent (NDA-99) modified by dimethylamine group as well as a nonionic macroporous adsorbent (XAD-4). The Langmuir and Freundlich isotherm models were employed to fit the experimental data to describe adsorption mechanism. It shows that NDA-99 resin exhibits an adsorption affinity for 2,4-D higher than XAD-4 resin owing to its exceptional micropore structure and the amine group of the hypercrosslinked matrix.Further studies indicate that the hydrogen bonding interaction and the stronger π-π conjugation play a significant role in the course of the adsorption of 2,4-D on NDA-99 resin, which is in agreement with the IR spectroscopic results and the △E values of HOMO (the highest occupied molecular orbit) of adsorbent and LUMO (the lowest unoccupied molecular orbit) of adsorbate calculated from the MINDO/3 model.     

An improved method of characterizing fatty acids by equivalent chain length values

A method has been developed for calculating more accurate and reproducible equivalent chain length (ECL) values of 33 straight-chain fatty acids (FA), with various positions and numbers of cis double bonds, chromatographed on DB-Wax and DB-1 columns. The dependence of ECL values on the position and the number of double bonds is described, as is a method of utilizing these relationships for the characterization of FA by gas chromatography.     

Interior Layers for Second-Order Singular Equations

Our aim in this article is to study singularly perturbed problems which display boundary layers in the interior of the domain. These interior boundary layers which supplement the usual boundary layers at the boundary, are generated by discontinuities in the data. Second-order linear elliptic one-dimensional and multi-dimensional problems are considered in this article.     

Principal Submatrices of a Hermitian Matrix

Suppose k 1 ,<, k m and n are positive integers such that k 1 + … + k m h n . We characterize those k i × k i Hermitian matrices A i , i = 1, < , m that can appear as diagonal blocks of an n × n Hermitian matrix C with prescribed eigenvalues. The characterization will be given in terms of the eigenvalues of C and A i , i = 1, <, m . Our results extend those of Thompson and Freede, Horn, Fan and Pall.     

Magneto-optical visualization of three spatial components of inhomogeneous stray fields

The article deals with the physical principles of magneto-optical visualization (MO) of three spatial components of inhomogeneous stray fields with the help of FeCo metal indicator films in the longitudinal Kerr effect geometry. The inhomogeneous field is created by permanent magnets. Both p- and s-polarization light is used for obtaining MO images with their subsequent summing, subtracting and digitizing. As a result, the MO images and corresponding intensity coordinate dependences reflecting the distributions of the horizontal and vertical magnetization components in pure form have been obtained. Modeling of both the magnetization distribution in the indicator film and the corresponding MO images shows that corresponding to polar sensitivity the intensity is proportional to the normal field component, which permits normal field component mapping. Corresponding to longitudinal sensitivity, the intensity of the MO images reflects the angular distribution of the planar field component. MO images have singular points in which the planar component is zero and their movement under an externally homogeneous planar field permits obtaining of additional information on the two planar components of the field under study. The intensity distribution character in the vicinity of sources and sinks (singular points) remains the same under different orientations of the light incidence plane. The change of incident plane orientation by π/2 alters the distribution pattern in the vicinity of the saddle points.     

A Singular Limit Problem for Conservation Laws Related to the Camassa–Holm Shallow Water Equation

We consider a shallow water equation of the Camassa–Holm type, which contains nonlinear dispersive effects as well as fourth order dissipative effects. We prove that as the diffusion and dispersion parameters tend to zero, with a condition on the relative balance between these two parameters, smooth solutions of the shallow water equation converge to discontinuous weak solutions of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the L ^p setting.