On the Levi and Lebesgue properties

The paper is devoted to a question if the Levi property is preserved by direct sums and quotients. The three-space problem for the Levi and Lebesgue properties in topological Riesz spaces is also investigated.     

Zeno Product Formula Revisited

We introduce a new product formula which combines an orthogonal projection with a complex function of a non-negative operator. Under certain assumptions on the complex function the strong convergence of the product formula is shown. Under more restrictive assumptions even operator-norm convergence is verified. The mentioned formula can be used to describe Zeno dynamics in the situation when the usual non-decay measurement is replaced by a particular generalized observables in the sense of Davies.     

Marcinkiewicz Integrals with Non-Doubling Measures

Let μ be a positive Radon measure on which may be non doubling. The only condition that μ must satisfy is μ(B(x, r)) ≤ Cr ⁿ for all , r > 0 and some fixed constants C > 0 and n ∈ (0, d]. In this paper, we introduce the Marcinkiewicz integral related to a such measure with kernel satisfying some H?rmander-type condition, and assume that it is bounded on L ²(μ). We then establish its boundedness, respectively, from the Lebesgue space L ¹(μ) to the weak Lebesgue space L ^1,∞(μ), from the Hardy space H ¹(μ) to L ¹(μ) and from the Lebesgue space L ^∞(μ) to the space RBLO(μ). As a corollary, we obtain the boundedness of the Marcinkiewicz integral in the Lebesgue space L ^p (μ) with p ∈ (1,∞). Moreover, we establish the boundedness of the commutator generated by the RBMO(μ) function and the Marcinkiewicz integral with kernel satisfying certain slightly stronger H?rmander-type condition, respectively, from L ^p (μ) with p ∈ (1,∞) to itself, from the space L log L(μ) to L ^1,∞(μ) and from H ¹(μ) to L ^1,∞(μ). Some of the results are also new even for the classical Marcinkiewicz integral. The third (corresponding) author was supported by National Science Foundation for Distinguished Young Scholars (No. 10425106) and NCET (No. 04-0142) of China.     

Some results on embeddings of algebras, after de Bruijn and McKenzie

In 1957, N.G. de Bruijn showed that the symmetric group Sym(Ω) on an infinite set Ω contains a free subgroup on 2^card(Ω) generators, and proved a more general statement, a sample consequence of which is that for any group A of cardinality card(Ω), the group Sym(Ω) contains a coproduct of 2^card(Ω) copies of A, not only in the variety of all groups, but in any variety of groups to which A belongs. His key lemma is here generalized to an arbitrary variety of algebras V, and formulated as a statement about functors Set V. From this one easily obtains analogs of the results stated above with “group” and Sym(Ω) replaced by “monoid” and the monoid Self(Ω) of endomaps of Ω, by “associative K-algebra” and the K-algebra End_K (V) of endomorphisms of a K-vector-space V with basis Ω, and by “lattice” and the lattice Equiv(Ω) of equivalence relations on Ω. It is also shown, extending another result from de Bruijn's 1957 paper, that each of Sym(Ω), Self(Ω) and End_K(V) contains a coproduct of 2^card(Ω) copies of itself.That paper also gave an example of a group of cardinality 2^card(Ω) that was not embeddable in Sym(Ω), and R. McKenzie subsequently established a large class of such examples. Those results are shown here to be instances of a general property of the lattice of solution sets in Sym(Ω) of sets of equations with constants in Sym(Ω). Again, similar results - this time of varying strengths - are obtained for Self(Ω), End_K(V), and Equiv(Ω), and also for the monoid Rel(Ω) of binary relations on Ω.Many open questions and areas for further investigation are noted.     

Lower bounds of some singular integrals and their applications

Let and . We are interested in the lower bounds of the integral:

On integer and fractional parts of powers of Salem numbers

Let N be a positive rational integer and let P be the set of powers of a Salem number of degree d. We prove that for any α∈P the fractional parts of the numbers , when n runs through the set of positive rational integers, are dense in the unit interval if and only if N≦ 2d − 4. We also show that for any α∈P the integer parts of the numbers αⁿ are divisible by N for infinitely many n if and only if N≦ 2d − 3. Received: 27 April 2005     

Product measures and convergence theorems for product integrals in integration theory for semigroup valued functions

In [11] Sion has investigated measures and integrals having values in uniform semigroups. In this paper a result which gives sufficient conditions for the existence of a unique product of semigroup valued measures is established. Moreover a convergence theorem for product integrals of semigroup valued functions is proved.     

Characterization of BCN films synthesized by radiofrequency plasma enhanced chemical vapor deposition

Boron carbonitride (BCN) films have been synthesized on Si(1 0 0) substrate by radio frequency plasma enhanced chemical vapor deposition using tris-(dimethylamino)borane (TDMAB) as a precursor. The deposition was performed at the different RF powers of 400-800 W, at the working pressure of 2×10⁻¹ Torr. The formation of the sp²-bonded BCN phase was confirmed by Fourier transform infrared spectroscopy. X-ray photoelectron spectroscopy measurements showed that B atoms were bonded to C and N atoms to form the BCN atomic hybrid configurations with the chemical compositions of B₅₂C₁₂N₃₆ (sample 1; prepared at the RF power of 400 W), B₅₂C₁₀N₃₈ (sample 2; at 500 W) and B₄₆C₁₈N₃₆ (sample 3; at 800 W), respectively. Near-edge X-ray absorption fine structure (NEXAFS) measurements indicated that B atoms were bonded not only to N atoms but also to C atoms to form various configurations of sp²-BCN atomic hybrids. The polarization dependence of NEXAFS suggested that the predominant hybrid configuration of sp²-BCN films oriented in the direction perpendicular to the Si substrate.     

Fluorescence characteristics of Dy/CdS nanocrystals doped silica xerogel

Cadmium sulfide semiconductor nanocrystals along with dysprosium ions were incorporated in silica xerogels through sol-gel route. The fluorescence properties were analyzed for samples annealed at different temperatures. The incorporation of the CdS semiconductors as nanocrystallites was confirmed from the HRTEM measurements. The fluorescence intensities were compared for Dy³⁺, CdS and Dy³⁺/CdS-doped silica xerogels. Enhanced blue emission was observed for codoped samples annealed up to 100 °C.     

Elastic behaviour of strontium-doped lanthanum calcium manganites in the vicinity of TC

Colossal magnetoresistance (CMR) manganites with compositional formula, La_0.67Ca_0.33−xSr_xMnO₃ (where x=0, 0.09, 0.11, 0.13, 0.33) were prepared by citrate gel route mainly to understand the elastic behaviour in the vicinity of their magnetic transition temperature T_C. The structural characterizations of all the materials clearly indicate that samples upto x=0.13 doping, are having orthorhombic structure with Pbnm space group and sample with x=0.33 is having rhombohedral structure with space group. The magnetic transition temperatures (T_C) are determined by AC susceptibility measurements and are found to increase continuously with increasing strontium concentration. Finally, a systematic investigation of the ultrasonic longitudinal velocities of all the samples was carried out. Further, all the samples are found to exhibit anomalous behaviour in the vicinity of their magnetic transition temperatures and the observed behaviour is explained using Landau's theory.