The uniform modulus of continuity of iterated Brownian motion

LetX be a Brownian motion defined on the line (withX(0)=0) and letY be an independent Brownian motion defined on the nonnegative real numbers. For allt0, we define theiterated Brownian motion (IBM),Z, by setting . In this paper we determine the exact uniform modulus of continuity of the process Z.Research supported by NSF grant DMS-9122242.     

Designing different nanomagnets via delocalization of cobalt magnetic moment inside narrow single walled carbon nanotubes

Density functional theory calculations were employed to study the effects of chirality and diameter of single walled carbon nanotubes (SWCNTs) on electronic, structural and magnetic properties of cobalt-doped (9,0), (5,5) and (5,0) nanotube systems. The (9,0) and (5,5) SWCNTs have similar diameters but different chiralities, whereas the (5,0) tube has a very small diameter. The Co-SWCNT systems are considered with four different possible arrangements, three of which are stable and only substitution of the Co with one of the carbon atoms on the surface of the SWCNTs is an exemption. Although the quasi-metallic band gap of the (9,0) SWCNT is eliminated by the cobalt doping process, metallic features of the (5,5) and (5,0) nanotubes remain unchanged. On the other hand, delocalization of the cobalt’s magnetization and inducement of a noticeable magnetization to the tubes provide a vast area of possible total magnetizations for the Co-SWCNT systems. The results are applicable to spintronics and useful for designing other nanomagnetic systems.     

Local times on curves and uniform invariance principles

Summary Sufficient conditions are given for a family of local times |L _t ^µ | ofd-dimensional Brownian motion to be jointly continuous as a function oft and . Then invariance principles are given for the weak convergence of local times of lattice valued random walks to the local times of Brownian motion, uniformly over a large family of measures. Applications included some new results for intersection local times for Brownian motions on ² and ².Research partially supported by NSF grant DMS-8822053     

Weak unimodality of finite measures, and an application to potential theory of additive Lévy processes

A probability measure on is called weakly unimodal if there exists a constant such that for all 0$">,

(0.1)

Here, denotes the -ball centered at with radius 0$">.

In this note, we derive a sufficient condition for weak unimodality of a measure on the Borel subsets of . In particular, we use this to prove that every symmetric infinitely divisible distribution is weakly unimodal. This result is then applied to improve some recent results of the authors on capacities and level sets of additive Lévy processes.

     

Kinetics and reaction mechanism of isothermal oxidation of Iranian ilmenite concentrate powder

Thermal oxidation of commercial ilmenite concentrate from Kahnouj titanium mines, Iran, at 500–950 °C was investigated for the first time. Fractional conversion was calculated from mass change of the samples during oxidation. Maximum FeO to Fe₂O₃ conversion of 98.63 % occurred at 900 °C after 120 min. Curve fit trials together with SEM line scan results indicated constant-size shrinking core model as the closest kinetic mechanism of the oxidation process. Below 750 °C, chemical reaction with activation energy of 80.65 kJ mol^?1 and between 775 and 950 °C, ash diffusion with activation energy of 53.50 kJ mol^?1 were the prevailing mechanisms. X-ray diffraction patterns approved presence of pseudobrookite, rutile, hematite, and Fe₂O₃·2TiO₂ phases after oxidation of ilmenite concentrate at 950 °C.     

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.

     

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.