Synthesis,Structural and Physicochemical Characterization of a Titanium(IV) Compound with the Hydroxamate Ligand N,2-Dihydroxybenzamide

The siderophore organic ligand N,2-dihydroxybenzamide (H₂dihybe) incorporates the hydroxamate group, in addition to the phenoxy group in the ortho-position and reveals a very rich coordination chemistry with potential applications in medicine, materials, and physical sciences. The reaction of H₂dihybe with TiCl₄ in methyl alcohol and KOH yielded the tetranuclear titanium oxo-cluster (TOC) [Ti^IV₄(μ-O)₂(HOCH₃)₄(μ-Hdihybe)₄(Hdihybe)₄]Cl₄∙10H₂O∙12CH₃OH (1). The titanium compound was characterized by single-crystal X-ray structure analysis, ESI-MS, ¹³C, and ¹H NMR spectroscopy, solid-state and solution UV–Vis, IR vibrational, and luminescence spectroscopies and molecular orbital calculations. The inorganic core Ti₄(μ-O)₂ of 1 constitutes a rare structural motif for discrete Ti^IV₄ oxo-clusters. High-resolution ESI-MS studies of 1 in methyl alcohol revealed the presence of isotopic distribution patterns which can be attributed to the tetranuclear clusters containing the inorganic core {Ti₄(μ-O)₂}. Solid-state IR spectroscopy of 1 showed the presence of an intense band at ~800 cm⁻¹ which is absent in the spectrum of the H₂dihybe and was attributed to the high-energy ν(Ti₂–μ-O) stretching mode. The ν(C=O) in 1 is red-shifted by ~10 cm⁻¹, while the ν(N-O) is blue-shifted by ~20 cm⁻¹ in comparison to H₂dihybe. Density Functional Theory (DFT) calculations reveal that in the experimental and theoretically predicted IR absorbance spectra of the ligand and Ti-complex, the main bands observed in the experimental spectra are also present in the calculated spectra supporting the proposed structural model. ¹H and ¹³C NMR solution (CD₃OD) studies of 1 reveal that it retains its integrity in CD₃OD. The observed NMR changes upon addition of base to a CD₃OD solution of 1, are due to an acid–base equilibrium and not a change in the Ti^IV coordination environment while the decrease in the complex’s lability is due to the improved electron-donating properties which arise from the ligand deprotonation. Luminescence spectroscopic studies of 1 in solution reveal a dual narrow luminescence at different excitation wavelengths. The TOC 1 exhibits a band-gap of 1.98 eV which renders it a promising candidate for photocatalytic investigations.     

On the Conditional Variance for Scale Mixtures of Normal Distributions

For a scale mixture of normal vector, X=A^1/2G, where X, G ⁿ and A is a positive variable, independent of the normal vector G, we obtain that the conditional variance covariance, Cov(X₂ X₁), is always finite a.s. for m2, where X₁ ⁿ and m<n, and remains a.s. finite even for m=1, if and only if the square root moment of the scale factor is finite. It is shown that the variance is not degenerate as in the Gaussian case, but depends upon a function S_A, m(·) for which various properties are derived. Application to a uniform and stable scale of normal distributions are also given.     

Prediction of stable processes: Spectral and moving average representations

Summary For stable processes which are Fourier transforms of processes with independent increments, we obtain a Wold decomposition, we characterize their regularity and singularity, and, in the discrete-parameter case, we derive their linear predictors. In sharp contrast with the Gaussian case, regular stable processes which are Fourier transforms of processes with independent increments are not moving averages of stable motion.This research was supported by the Air Force Office of Scientific Research Grant No. F 49620 82C0009     

Representation of stochastic processes of second order and linear operations

Series representations are obtained for the entire class of measurable, second order stochastic processes defined on any interval of the real line. They include as particular cases all earlier representations; they suggest a notion of “smoothness” that generalizes well-known continuity notions; and they decompose the stochastic process into two orthogonal parts, the smooth part and a strongly discontinuous part. Also linear operations on measurable, second order processes are studied; it is shown that all “smooth” linear operations on a process, and, in particular, all linear operations on a “smooth” process, can be approximated arbitrarily closely by linear operations on the sample paths of the process.     

Positive Gegenbauer Polynomial Sums and Applications to Starlike Functions

Let $s_n(f,z):=\sum_{k=0}^{n}a_kz^k$ be the $n$th partial sum of $f(z)=\sum_{k=0}^{\infty{}}a_kz^k$. We show that $\RE s_n(f/z,z)>0$ holds for all $z\in\D,\ n\in\N$, and all starlike functions $f$ of order $\lambda$ iff $\lambda_0\leq\lambda<1$ where $\lambda_0=0.654222...$ is the unique solution $\lambda\in(\frac{1}{2},1)$ of the equation $\int_{0}^{3\pi/2}t^{1-2\lambda}\cos t \,dt=0$. Here $\D$ denotes the unit disk in the complex plane $\C$. This result is the best possible with respect to $\lambda_0$. In particular, it shows that for the Gegenbauer polynomials $C_{n}^{\mu}(x)$ we have $\sum_{k=0}^n C_{k}^{\mu}(x)\cos k \theta>0$ for all $n\in\N,\ x\in[-1,1]$, and $0<\mu\leq\mu_0:=1-\lambda_0=0.345778...$. This result complements an inequality of Brown, Wang, and Wilson (1993) and extends a result of Ruscheweyh and Salinas (2000).     

On α-symmetric multivariate distributions

A random vector is said to have a 1-symmetric distribution if its characteristic function is of the form φ(|t₁| + … + |t_n|). 1-Symmetric distributions are characterized through representations of the admissible functions φ and through stochastic representations of the radom vectors, and some of their properties are studied.     

On the theory of elliptically contoured distributions

The theory of elliptically contoured distributions is presented in an unrestricted setting, with no moment restrictions or assumptions of absolute continuity. These distributions are defined parametrically through their characteristic functions and then studied primarily through the use of stochastic representations which naturally follow from the work of Schoenberg [5] on spherically symmetric distributions. It is shown that the conditional distributions of elliptically contoured distributions are elliptically contoured, and the conditional distributions are precisely identified. In addition, a number of the properties of normal distributions (which constitute a type of elliptically contoured distributions) are shown, in fact, to characterize normality.