Symmetric operators with real defect subspaces of the maximal dimension. Applications to differential operators

Let H be a Hilbert space and let A be a simple symmetric operator in H with equal deficiency indices d:=n_±(A)<∞. We show that if, for all λ in an open interval I⊂R, the dimension of defect subspaces Nλ(A) (=Ker(A^?−λ)) coincides with d, then every self-adjoint extension has no continuous spectrum in I and the point spectrum of is nowhere dense in I. Application of this statement to differential operators makes it possible to generalize the known results by Weidmann to the case of an ordinary differential expression with both singular endpoints and arbitrary equal deficiency indices of the minimal operator.     

Crystal structure and morphology evolution in the LaXO3, X = Al, Ga, In nano-oxide series. Consequences for the synthesis of luminescent phosphors

The LaXO(3):Tb(3+) (X = Al(3+), Ga(3+), In(3+)) perovskite nanoparticles were obtained using the nonhydrolytic treatment (Bradley reaction) of the molecular precursors of the La(O(i)Pr)(3), Al(O(i)Pr)(3), Ga(O(i)Pr)(3), In(5)O(O(i)Pr)(13), and Tb(acac)(3), respectively. It was shown that crystal structure and morphology evolution in the LaXO(3), X = Al, Ga, In nano-oxide series depended on the size and chemical properties of the X-metal atom. Formation of the LaInO(3):Tb(3+) nanoparticles is distinctly less thermodynamically demanding on contrary to the LaAlO(3):Tb(3+) and LaGaO(3):Tb(3+) since it provided crystalline product directly in the solution synthesis at 202 °C, which is the lowest reported synthesis temperature for this compound up-to-date. This behavior was ascribed to the effects directly connected with the dopant substitution (exchange of bigger La(3+) cation with smaller Tb(3+)) as well as reduction of the particle size. The size effects are mostly reflected in the expansion of the cell volume, changes of the cell parameters as well as shifting and broadening of the Raman bands. Indirectly, size reduction has also an effect on the luminescence properties through the higher probability of presence of surface and net defects as well as heterogeneous distribution of the Tb(3+) ions caused by high surface-to-volume ratio. The prepared nanophosphors show basically green emission with exception of white-green in case of the LaInO(3):Tb(3+). Strong emission quenching was found in the latter case being most likely a consequence of the nonradiative energy transfer between Tb(3+) and In(3+) as well as the presence of defects. In comparison to the Pechini's method, the LaXO(3) nanoparticles required significantly lower annealing temperature (700 °C) necessary for complete crystallization. Generally the resulting particles are distinctly smaller (5 to 25 nm) and less agglomerated (50-100 nm) depending on the reaction conditions as well as thermal treatment. For the first time, it was shown that the LaGaO(3):Tb(3+) nanopowder has crystallized in the high-temperature rhombohedral R3c phase.     

Controlling phase formation in solids: rational synthesis of phase separated Co@Fe2O3 heteroparticles and CoFe2O4 nanoparticles

A wet chemical approach from organometallic reactants allowed the targeted synthesis of Co@Fe(2)O(3) heterodimer and CoFe(2)O(4) ferrite nanoparticles. They display magnetic properties that are useful for magnetic MRI detection.     

A Raman microprobe study of melt inclusions in kimberlites from Siberia, Canada, SW Greenland and South Africa

The geometries, energies and vibrational frequencies of various polyborates in both gaseous and aqueous phase were calculated at the B3LYP/aug-cc-pVDZ level. The calculated total symmetrical stretching Raman shifts of B(OH)(3), B(OH)(4)(-), B(2)O(OH)(4), B(2)O(OH)(5)(-), B(2)O(OH)(6)(2-), B(3)O(3)(OH)(3), B(3)O(3)(OH)(4)(-), B(3)O(3)(OH)(5)(2-), B(3)O(3)(OH)(6)(3-), B(4)O(5)(OH)(4)(2-) and B(5)O(6)(OH)(4)(-) were assigned to 877.40, 735.33, 785.22, 792.90, 696.79, 587.72, 599.06, 740.16, 705.01, 551.67 and 521.04cm(-1), respectively. The results can be used as the characteristic frequency for polyborates in aqueous phase at room temperature. At least six types of polyborates B(OH)(3), B(OH)(4)(-), B(3)O(3)(OH)(4)(-), B(3)O(3)(OH)(5)(2-), B(4)O(5)(OH)(4)(2-) and B(5)O(6)(OH)(4)(-), occur in aqueous solutions at ambient temperature. The chemical species distribution and the relevant interaction mechanisms among polyborates in the solutions were also suggested.     

Theoretical investigations into the enantiomeric and racemic forms of α-(trifluoromethyl)lactic acid

There are many different hypotheses on the origin of biomolecular homochirality. One possible scenario concerns the enantiomeric enrichment of a nearly racemic solid via self-disproportionation of enantiomers. In particular, in a recent paper Soloshonok and co-workers showed a first example of optical self-purification of α-(trifluoromethyl)lactic acid by sublimation [V. A. Soloshonok et al., J. Am. Chem. Soc. 2007, 129, 12112]. Here we present detailed theoretical studies of α-(trifluoromethyl)lactic acid in the solid state as well as in the gas-phase dimeric form. The calculations of energy differences between dimers show that in the solid state the enantiomeric pure compound is energetically preferred, while in the gas phase the equilibrium shifts towards the racemic mixture although thermodynamic corrections cannot be neglected, thus providing a detailed microscopic explanation for the enantio-purification process for the first time.     

Solutions of KZ differential equations modulo p

The Ramanujan Journal - We construct polynomial solutions of the KZ differential equations over a finite field $${\mathbb F}_p$$ as analogs of hypergeometric solutions.     

Spherically symmetric random permutations

We consider random permutations which are spherically symmetric with respect to a metric on the symmetric group S_n and are consistent as n varies. The extreme infinitely spherically symmetric permutation‐valued processes are identified for the Hamming, Kendall‐tau and Cayley metrics. The proofs in all three cases are based on a unified approach through stochastic monotonicity.     

The Subadditive Ergodic Theorem and generic stretching factors for free group automorphisms

Let F _k be a free group of rank k ≥ 2 with a fixed set of free generators. We associate to any homomorphism φ from F _k to a group G with a left-invariant semi-norm a generic stretching factor, λ(φ), which is a noncommutative generalization of the translation number. We concentrate on the situation where φ: F _k → Aut(X) corresponds to a free action of F _k on a simplicial tree X, in particular, where φ corresponds to the action of F _k on its Cayley graph via an automorphism of F _k . In this case we are able to obtain some detailed “arithmetic” information about the possible values of λ = λ(φ). We show that λ ≥ 1 and is a rational number with 2kλ ∈ ℤ[1/(2k − 1)] for every φ ∈ Aut(F _k ). We also prove that the set of all λ(φ), where φ varies over Aut(F _k ), has a gap between 1 and 1+(2k−3)/(2k ²−k), and the value 1 is attained only for “trivial” reasons. Furthermore, there is an algorithm which, when given φ, calculates λ(φ). The second and the third author were supported by the NSF grant DMS#0404991 and the NSA grant DMA#H98230-04-1-0115.     

Inversionless Amplification in a Three-Level System Driven by a Single Field

The three-level system subjected to electromagnetic fields is investigated. It is shown that when one of two fields is strong (the pumping field), the system can amplify the second (probe) field even if no population inversion is created, but some phase relations are satisfied. The conditions necessary for inversionless amplification to take place are discussed. The connection between this effect and stimulated Raman emission is shown. Numerical simulations for each one of the discussed cases were performed.