Hausdorff dimension of the contours of symmetric additive Lévy processes

Let X ₁, ..., X _N denote N independent, symmetric Lévy processes on R ^d . The corresponding additive Lévy process is defined as the following N-parameter random field on R ^d : Khoshnevisan and Xiao (Ann Probab 30(1):62–100, 2002) have found a necessary and sufficient condition for the zero-set of to be non-trivial with positive probability. They also provide bounds for the Hausdorff dimension of which hold with positive probability in the case that can be non-void. Here we prove that the Hausdorff dimension of is a constant almost surely on the event . Moreover, we derive a formula for the said constant. This portion of our work extends the well known formulas of Horowitz (Israel J Math 6:176–182, 1968) and Hawkes (J Lond Math Soc 8:517–525, 1974) both of which hold for one-parameter Lévy processes. More generally, we prove that for every nonrandom Borel set F in (0,∞) ^N , the Hausdorff dimension of is a constant almost surely on the event . This constant is computed explicitly in many cases. The research of N.-R. S. was supported by a grant from the Taiwan NSC.     

Packing dimension of the range of a Lévy process

Let denote a Lévy process in with exponent . Taylor (1986) proved that the packing dimension of the range is given by the index

We provide an alternative formulation of in terms of the Lévy exponent . Our formulation, as well as methods, are Fourier-analytic, and rely on the properties of the Cauchy transform. We show, through examples, some applications of our formula.

     

A simple route to synthesize nanocrystalline nickel ferrite (NiFe2O4) in the presence of octanoic acid as a surfactant

Nanocrystalline nickel ferrite (NiFe₂O₄) powder was prepared by a co-precipitation method from Ni and Fe chlorides. The as-prepared samples were characterized by powder X-ray diffractometry (XRD), transmission electron microscopy (TEM), scanning electron microscopy (SEM), Fourier transform infrared spectroscopy (FT-IR) and vibrating sample magnetometry (VSM). SEM and TEM indicated that the particles were spherical with particle sizes in the range 25 ± 5 nm. The magnetic properties of the sample were measured by using a vibrating sample magnetometer, which showed that the sample exhibited typical ferromagnetic behavior at room temperature, while a finite coercivity of 245.5 Oe was present at 300 K. The saturation magnetization of the sample (23.13 emu/g) was significantly lower than that for the reported multidomain bulk particles (55 emu/g), reflecting the ultrafine nature of the sample.     

Sonochemical synthesis of Dy2(CO3)3 nanoparticles,Dy(OH)3 nanotubes and their conversion to Dy2O3 nanoparticles

Dysprosium carbonates nanoparticles were synthesized by the reaction of dysprosium acetate and NaHCO₃ by a sonochemical method. Dysprosium oxide nanoparticles with average size about 17 nm were prepared from calcination of Dy₂(CO₃)₃·1.7H₂O nanoparticles. Dy(OH)₃ nanotubes were synthesized by sonication of Dy(OAC)₃·6H₂O and N₂H₄. The as-synthesized nanostructures were characterized by scanning electron microscopy (SEM), X-ray powder diffraction (XRD), transmission electron microscopy (TEM), and Fourier transform infrared spectroscopy (FT-IR). Photoluminescence measurement shows that the nanoparticles have two emission peaks around 17,540 cm^?1 and 20,700 cm^?1, which should come from the electron transition from ⁴F_9/2 → ⁶H_15/2 levels and ⁴F_9/2 → ⁶H_13/2 levels, respectively. The effect of calcination temperature and sonication time was investigated on the morphology and particle size of the products. The sizes could be controlled by the feeding rate of the precipitating agent (NaHCO₃ and N₂H₄) and slower feeding rate lead to smaller nanoparticles.     

Solvent effects on chemical processes. 3. Surface tension of binary aqueous organic solvents

The surface tension of binary solvents is modelled by analogy to solvation effects arising from solvent-solute interactions. Competitive exchange equilibria are postulated between solvent component I (water) and solvent component 2 (organic cosolvent) for solute, which in this case is air; the solvation shell is thus the surface phase. A quantitative relationship is given between surface tension and mole fractions x₁ and x₂, the model parameters being exchange equilibrium constants K₁ and K₂. The equation is analyzed, it is applied to literature surface tension data, and it is compared with an earlier model from this laboratory. Curve-fits are very good, and the parameters appear to possess physical significance.     

Decorrelation of Total Mass Via Energy

The main result of this small note is a quantified version of the assertion that if u and v solve two nonlinear stochastic heat equations, and if the mutual energy between the initial states of the two stochastic PDEs is small, then the total masses of the two systems are nearly uncorrelated for a very long time. One of the consequences of this fact is that a stochastic heat equation with regular coefficients is a finite system if and only if the initial state is integrable.     

Controllable Synthesis of Covellite Nanoparticles via Thermal Decomposition Method

In this study, covellite (CuS) nanoparticles were synthesized through a facile and low temperature thermal decomposition method using [Cu(sal)₂]- oleylamine complex, (sal = salicylaldehydeato, prepared in situ from [Cu(sal)₂] and oleylamine as the precursors), and sulfur as the Cu²⁺ source and S source, respectively. Scanning electron microscope, transmission electron microscope, electron diffraction and ultraviolet–visible absorption (UV–Vis) spectra were used for the characterization of the products. The effect of reaction parameters, such as the copper:sulfur molar ratio, the reaction temperature and the reaction time on the shape, size and phase of CuS nanostructures, was investigated. The results showed that the, covellite (hexagonal structure of CuS) with an average size between 20 and 45 nm could be obtained with the Cu:S molar ratio of 1: 3 at 105 °C for 60 min. With increasing the reaction temperature from 105 to 200 °C, non-stoichiometric Cu_1.65S with the average size of 25–50 nm was obtained due to the different existing state of the released Cu²⁺ ions from the copper-oleylamine complex.     

Synthesis and characterization of cobalt sulfide nanocrystals in the presence of thioglycolic acid via a simple hydrothermal method

Co₃S₄ nanocrystals were synthesized via a hydrothermal technique by adding thioglycolic acid and Co(CH₃COO)₂·4H₂O as the precursors. The effect of some parameters, such as the reaction time and temperature and concentration of reactants, on the growth and morphology of the nano-structures has been studied. The products were characterized by X-ray diffraction (XRD), scanning electron microscopy (SEM), photoluminescence (PL) spectroscopy and Fourier transform infrared (FT-IR) spectra.     

Synthesis and characterization of spinel-type CuAl<Subscript>2</Subscript>O<Subscript>4</Subscript> nanocrystalline by modified sol–gel method

Nanocrystalline Copper aluminate (CuAl₂O₄) was prepared by sol–gel technique using aluminum nitrate, copper nitrate, diethylene glycol monoethyl ether and citric acid were used as precursor materials. This method starts from of the precursor complex, and involves formation of homogeneous solid intermediates, reducing atomic diffusion processes during thermal treatment. The formation of pure crystallized CuAl₂O₄ nanocrystals occurred when the precursor was heat-treated at 600 °C in air for 2 h. The stages of the formation of CuAl₂O₄, as well as the characterization of the resulting compounds were done using thermo–gravimetric analysis, X-ray diffraction, scanning electron microscopy and Fourier transform infrared spectroscopy. The products were analyzed by transmission electron microscopy and ultraviolet–visible (UV–Vis) spectroscopy to be round, about 17–26 nm in size and E _g = 2.10 eV.     

On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

We study the stochastic heat equation ${\partial_t u = \mathcal{L}u+\sigma(u)\dot W}$ in (1?+?1) dimensions, where ${\dot W}$ is space-time white noise, σ : R → R is Lipschitz continuous, and ${\mathcal{L}}$ is the generator of a symmetric Lévy process that has finite exponential moments, and u ₀ has exponential decay at ±∞. We prove that under natural conditions on σ : (i) The νth absolute moment of the solution to our stochastic heat equation grows exponentially with time; and (ii) The distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. Very little else seems to be known about the location of the high peaks of the solution to the stochastic heat equation under the present setting (see, however, G?rtner et?al. in Probab Theory Relat Fields 111:17–55, 1998; G?rtner et?al. in Ann Probab 35:439–499, 2007 for the analysis of the location of the peaks in a different model). Finally, we show that these results extend to the stochastic wave equation driven by Laplacian.