Local asymptotic laws for the Brownian convex hull

Summary We prove a general theorem for the precise rate at which the convex hull of Brownian motion gets created. The latter result relates large deviation theory to P. Lévy's geometric proof of Strassen's law of the iterated logarithm. This also answers a question of S. Evans. Moreover, we give a partial solution to a question of J. Hammersley and P. Lévy regarding the slowness of the growth of the hull process. Several examples, some classical and some new, are given to illustrate the theorems. Finally, we present applications to the convex hull of random walks ind dimensions.     

On the chaotic character of the stochastic heat equation, II

Consider the stochastic heat equation ${\partial_t u = (\varkappa/2)\Delta u+\sigma(u)\dot{F}}$ , where the solution u := u _t (x) is indexed by ${(t, x) \in (0, \infty) \times {\bf R}^d}$ , and ${\dot{F}}$ is a centered Gaussian noise that is white in time and has spatially-correlated coordinates. We analyze the large- ${\|x\|}$ fixed-t behavior of the solution u in different regimes, thereby study the effect of noise on the solution in various cases. Among other things, we show that if the spatial correlation function f of the noise is of Riesz type, that is ${f(x)\propto \|x\|^{-\alpha}}$ , then the “fluctuation exponents” of the solution are ${\psi}$ for the spatial variable and ${2\psi-1}$ for the time variable, where ${\psi:=2/(4-\alpha)}$ . Moreover, these exponent relations hold as long as ${\alpha \in (0, d \wedge 2)}$ ; that is precisely when Dalang’s theory [Dalang, Electron J Probab 4:(Paper no. 6):29, 1999] implies the existence of a solution to our stochastic PDE. These findings bolster earlier physical predictions [Kardar et al., Phys Rev Lett 58(20):889–892, 1985; Kardar and Zhang, Phys Rev Lett 58(20):2087–2090, 1987].     

Simple Hydrothermal Synthesis of Nickel Hydroxide Flower-Like Nanostructures

α-Ni(OH)₂ flower-like nanostructures were successfully synthesized through one-step hydrothermal method with nickel acetate tetrahydrate, ethylene-1,2-diamine (en), hexamethylenetetramine (HMT) and cetyltrimethylammonium bromide (CTAB) as morphology-directing agents. Optimum conditions to obtain high yield and pure phase α-Ni(OH)₂ were identified by varying experimental parameters such as: en, HMT and CTAB concentration and reaction temperature. The products were characterized by X-ray diffraction, scanning electron microscopy, Fourier transform infrared and thermogravimetric analysis. These results indicated that the α-nickel hydroxide contains water molecules and anions. The hierarchical NiO nanostructures were obtained by the as-synthesized α-Ni(OH)₂ nanostructures annealed at 300 °C for 4 h.     

Dynamical percolation on general trees

Häggström et al. (Ann Inst H Poincaré Probab Stat 33(4):497–528, 1997) have introduced a dynamical version of percolation on a graph G. When G is a tree they derived a necessary and sufficient condition for percolation to exist at some time t. In the case that G is a spherically symmetric tree (Peres and Steif in Probab Theory Relat Fields 111(1):141–165, 1998), derived a necessary and sufficient condition for percolation to exist at some time t in a given target set D. The main result of the present paper is a necessary and sufficient condition for the existence of percolation, at some time ${t\in D}H?ggstr?m et al. (Ann Inst H Poincaré Probab Stat 33(4):497–528, 1997) have introduced a dynamical version of percolation on a graph G. When G is a tree they derived a necessary and sufficient condition for percolation to exist at some time t. In the case that G is a spherically symmetric tree (Peres and Steif in Probab Theory Relat Fields 111(1):141–165, 1998), derived a necessary and sufficient condition for percolation to exist at some time t in a given target set D. The main result of the present paper is a necessary and sufficient condition for the existence of percolation, at some time , in the case that the underlying tree is not necessary spherically symmetric. This answers a question of Yuval Peres (personal communication). We present also a formula for the Hausdorff dimension of the set of exceptional times of percolation. Research supported in part by a grant from the National Science Foundation.     

Intersections of Brownian motions

This article presents a survey of the theory of the intersections of Brownian motion paths. Among other things, we present a truly elementary proof of a classical theorem of A. Dvoretzky, P. Erdős and S. Kakutani. This proof is motivated by old ideas of P. Lévy that were originally used to investigate the curve of planar Brownian motion.     

Shape control of nickel selenides synthesized by a simple hydrothermal reduction process

Nickel selenide (NiSe) nanoparticles were prepared from the reaction of a SeCl₄ aqueous solution with a NiCl₂·6H₂O aqueous solution in the presence of polyvinyl alcohol (PVA) as a capping agent and hydrazine hydrate (N₂H₄·H₂O) as a reductant through a hydrothermal method. The size, morphology, chemical composition and purity of these nanoparticles depend on the capping agent, reductant, reaction temperature and time.     

Charge transfer complexes of adenosine-5'-monophosphate and cytidine-5'-monophosphate with water-soluble cobalt(II) Schiff base complexes in aqueous solution

Water-soluble cobalt(II) tetradentate Schiff base complexes have been shown to form charge transfer (CT) complexes with a series of nucleoside monophosphates including adenosine-5'-monophosphate (AMP) and cytidine-5'-monophosphate (CMP). The investigated water-soluble cobalt(II) Schiff base complexes are (i) disodium[{bis(5-sulfo-salicylaldehyde)-o-phenylenediiminato}cobalt(II)], Na2[Co(SO3-salophen)] (1); (ii) disodium[{bis(5-sulfo-salicylaldehyde)-4,5-dimethyl-o-phenylenediiminato}cobalt(II)], Na2[Co(SO3-sal-4,5-dmophen)] (2) and (iii) disodium[{bis(4-methoxy-5-sulfo-salicylaldehyde)-4,5-dimethyl-o-phenylenediiminato}cobalt(II)], Na(2)[Co(SO3-4-meosal-4,5-dmophen)] (3). The formation constant and thermodynamic parameters for charge transfer complex formation of water-soluble cobalt(II) Schiff base complexes with nucleoside monophosphates were determined spectrophotometrically in aqueous solution at constant ionic strength (I = 0.2 mol dm(-3) KNO3) under physiological condition (pH 7.0) and at various temperatures between 288 and 308 K. The stoichiometry has been found to be 1:1 (water-soluble cobalt(II) Schiff base complex: nucleoside monophosphate) in each case. Our spectroscopic and thermodynamic results show that the interaction of water-soluble cobalt(II) Schiff base complexes with the investigated nucleoside monophosphates occurs mainly through the phosphate group. The trend of the interaction according to the cobalt(II) Schiff base complexes due to electronic and steric factors is as follows: Na2[Co(SO3-salophen)] > Na2[Co(SO3-sal-4,5-dmophen)] > Na2[Co(SO3-4-meosal-4,5-dmophen)]. Also the trend of the interaction of a given cobalt(II) Schiff base complex according to the nucleoside monophosphate is as follows: CMP > AMP.     

Exceptional times and invariance for dynamical random walks

Consider a sequence of i.i.d. random variables. Associate to each X _i (0) an independent mean-one Poisson clock. Every time a clock rings replace that X-variable by an independent copy and restart the clock. In this way, we obtain i.i.d. stationary processes {X _i (t)} _{t ≥0} (i=1,2,···) whose invariant distribution is the law ν of X ₁(0). Benjamini et al. (2003) introduced the dynamical walk S _n (t)=X ₁(t)+···+X _n (t), and proved among other things that the LIL holds for n↦S _n (t) for all t. In other words, the LIL is dynamically stable. Subsequently (2004b), we showed that in the case that the X _i (0)'s are standard normal, the classical integral test is not dynamically stable. Presently, we study the set of times t when n↦S _n (t) exceeds a given envelope infinitely often. Our analysis is made possible thanks to a connection to the Kolmogorov ɛ-entropy. When used in conjunction with the invariance principle of this paper, this connection has other interesting by-products some of which we relate. We prove also that the infinite-dimensional process converges weakly in to the Ornstein–Uhlenbeck process in For this we assume only that the increments have mean zero and variance one. In addition, we extend a result of Benjamini et al. (2003) by proving that if the X _i (0)'s are lattice, mean-zero variance-one, and possess (2+ɛ) finite absolute moments for some ɛ>0, then the recurrence of the origin is dynamically stable. To prove this we derive a gambler's ruin estimate that is valid for all lattice random walks that have mean zero and finite variance. We believe the latter may be of independent interest. The research of D. Kh. is partially supported by a grant from the NSF.     

Synthesis,characterization, crystal structure and oxygen-evolution activity of a manganese(II) complex with 2,4,6-tris (2-pyridyl)-1,3,5-triazine

The title complex [Mn(tptz)(CH₃COO)(OH₂)₂]NO₃ was synthesized through the reaction of tptz (2,4,6-tris(2-pyridyl)-1,3,5-triazine), nitric acid and manganese(II) acetate. The molecular structure was characterized by X-ray diffraction, elemental analysis, electrochemistry, EPR, IR, fluorescence and UV–Vis spectroscopy and its oxygen evolving activity has been studied. X-ray structure analysis shows that each Mn(II) ion is seven coordinated by a bidentate acetate, three nitrogen atoms of tptz and two oxygen atoms from two water ligands, which are coordinated in axial positions. The complex acts as an oxygen evolving complex with oxone (2KHSO₅·KHSO₄·K₂SO₄) as primary oxidant in aqueous solution with a turnover number of 1 (mol of O₂/mol of the complex). Kinetic studies revealed a first-order dependence on the complex and oxidant. The EPR spectrum shows that the mononuclear complex oxidize to a Mn^III,IV₂ di-μ-oxo by oxone.     

A novel chelating acid-assisted thermolysis procedure for preparation of tin oxide nanoparticles

Pure tin dioxide (SnO₂) nanoparticles were synthesized via thermolysis of tin phthalate and tin oxalate in the presence of oleic acid (OA) as solvent. Oleic acid (OA) was employed as an organic solvent, which can be applied to control particle growth and to stabilize the particles. The products were characterized by X-ray diffraction (XRD), scanning electron microscopy (SEM), transmission electron microscopy (TEM), Fourier transform infrared (FT-IR) spectroscopy, X-ray photoelectron spectroscopy (XPS) and photoluminescence (PL) spectroscopy. The orthorhombic phase SnO₂ nanoparticles with average size about 12 nm were synthesized through thermolysis of tin phthalate in the presence of oleic acid.