The existence of a random attractor in H1(R3)×L2(R3) is proved for the damped semilinear stochastic wave equation defined on the entire space R 3. The nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent. The uniform pullback estimates on the tails of solutions for large space variables are established. The pullback asymptotic compactness of the rand...

The self intersection of an immersion i : S2 → R3 dissects S2 into pieces which are planar surfaces (unless i is an embedding). In this work we determine what collections of planar surfaces may be obtained in this way. In particular, for every n we construct an immersion i: S2 → R3 with 2n triple points, for which all pieces are discs. Keyword Immersions of the 2-sphere Mathematics Subject Clas...

In the avian hindbrain, the loss of premigratory neural crest cells from rhombomeres 3 and 5 (r3, r5) through programmed cell death contributes to the patterning of emigrant crest cells into three discrete streams. Programmed cell death is induced by the upregulation of Bmp4 and Msx2 in r3 and r5. We show that cSFRP2, a WNT antagonist, is expressed in the even-numbered rhombomeres and that over...

Let Aq (α, α, k) be the scattering amplitude, corresponding to a local potential q(x), x ∈ R3, q(x) = 0 for |x | > a, where a > 0 is an arbitrary large fixed number, α, α ∈ S2 are unit vectors, S2 is the unit sphere in R3, α is the direction of the incident wave, k2 > 0 is the energy. We prove that given an arbitrary function f (α) ∈ L2(S2), an arbitrary fixed α0 ∈ S 2, an arbitrary fixed k > 0...

Transmission electron microscopy (TEM) based orientation mapping has been used to measure the length fraction of coherent and incoherent R3 grain boundaries in a series of six nanocrystalline Cu thin films with thicknesses in the range of 26–111 nm and grain sizes from 51 to 315 nm. The films were annealed at the same temperature (600 °C) for the same length of time (30 min), have random textur...

We construct a cubical CW-complex CK(M3) whose rational cohomology algebra contains Vassiliev invariants of knots in the 3-manifold M3. We construct CK(R3) by attaching cells to CK(R3) for every degenerate 1-singular and 2-singular knot, and we show that π1(CK(R 3)) = 1 and π2(CK(R 3)) = Z. We give conditions for Vassiliev invariants to be nontrivial in cohomology. In particular, for R3 we show...

One electron system minimally coupled to a quantized radiation field is considered. It is assumed that the quantized radiation field is massless, and no infrared cutoff is imposed. The Hamiltonian, H, of this system is defined as a self-adjoint operator acting on L2(R3) ⊗ F , where F is the Boson Fock space over L2(R3 × {1, 2}). It is shown that the ground state, ψg, of H belongs to ∩k=1D(1 ⊗Nk...

Four identical 37.5 Liter UASB reactors R1, R2, R3 and R4 were used to study the effect of sulfate concentration on granule formation. Diluted molasses with COD range 1000–1300 mg/l were used as feed and acclimated cow manure was used as seed. Concentration of sulfate ions in the four reactors were 100, 500, 1000 and 1500mg/l. Granules were observed in R2 after 33 days from startup time, while ...