On Krein’s Differential System and its Generalization

In this article we investigate Krein’s differential systems as well as correct some assertions both in M.G. Krein’s article and in our works dedicated to the Krein’s systems and their generalization.     

On-line Tracking of a Smooth Regression Function

We construct an on-line estimator with equidistant design for tracking a smooth function from Stone–Ibragimov–Khasminskii’s class. This estimator has the optimal convergence rate of risk to zero in sample size. The procedure for setting coefficients of the estimator is controlled by a single parameter and has a simple numerical solution. The off-line version of this estimator allows to eliminate a boundary layer. Simulation results are given. This work is partially supported by a fellowship from the Yitzhak and Chaya Weinstein Research Institute for Signal Processing at Tel Aviv University.     

A kinetic model for step flow growth in molecular beam epitaxy

A terrace-step-kink model for epitaxial step flow growth of steps with no bonds along them is derived from kinetic arguments. The model is combined with an existing model for the steps that have strong bonding along them to describe steps of arbitrary orientation in terms of densities of adatoms, step adatoms and kinks. A planar steady-state solution for a simplified version of the model is constructed and analyzed. Different mass transport mechanisms are modeled that result in different far-from-equilibrium behavior, confirming that edge diffusion is the main factor stabilizing the steps during growth. Furthermore kinetic Wulff shapes are constructed from the calculated step velocities.     

A high-sensitivity photometric method based on doped Sol-Gel glass detectors: determination of sub-ppb divalent iron

Summary A novel high-sensitivity photometric method to determine trace concentrations of water pollutants is presented. The Sol-Gel procedure, a simple and efficient way to produce porous glasses at room temperature, is applied to entrap organic reagents (e.g. 1,10-phenanthroline) in glass matrices without resorting to complicated covalent bonding procedures. When these doped Sol-Gel glass detectors are immersed in water they concentrate the analyte [e.g. iron(II)] from the solution into the glass matrix. The glass detectors may then be analyzed by a conventional spectrophotometer. Thus, exceedingly low concentrations can be determined, e.g., the demonstrated detection limit of ferrous ions in aqueous solution is below 100 ppt.     

Progression-free sets in finite abelian groups

Let G be a finite abelian group. Write and denote by rk(2G) the rank of the group 2G.Extending a result of Meshulam, we prove the following. Suppose that A⊆G is free of “true” arithmetic progressions; that is, a₁+a₃=2a₂ with a₁,a₂,a₃∈A implies that a₁=a₃. Then |A|<2|G|/rk(2G). When G is of odd order this reduces to the original result of Meshulam.As a corollary, we generalize a result of Alon and show that if an integer k?2 and a real ε>0 are fixed, |2G| is large enough, and a subset A⊆G satisfies |A|?(1/k+ε)|G|, then there exists A₀⊆A such that 1?|A₀|?k and the elements of A₀ add up to zero. When G is of odd order or cyclic this reduces to the original result of Alon.     

The Århus integral of rational homology 3-spheres III: Relation with the Le–Murakami–Ohtsuki invariant

Continuing the work started in [Å-I] and [Å-II], we prove the relationship between the Århus integral and the invariant (henceforth called LMO) defined by T.Q.T. Le, J. Murakami and T. Ohtsuki in [LMO]. The basic reason for the relationship is that both constructions afford an interpretation as integrated holonomies. In the case of the Århus integral, this interpretation was the basis for everything we did in [Å-I] and [Å-II]. The main tool we used was formal Gaussian integration. For the case of the LMO invariant, we develop an interpretation of a key ingredient, the map j_m, as formal negative dimensional integration. The relation between the two constructions is then an immediate corollary of the relationship between the two integration theories.     

Binuclear copper(I) macrocycles synthesized via the weak-link approach

The weak-link approach has been employed to synthesize a series of bimetallic Cu(I) macrocycles in high yield. Addition of phosphinoalkylether or -thioether ligands to [Cu(MeCN)4]PF6 produces "condensed" intermediates, [mu-(1,4-(PPh2CH2CH2X)2Y)2Cu2][PF6]2 (X = S, O; Y = C6H4, C6F4), containing strong P-Cu bonds and weaker O-Cu or S-Cu bonds. The weak bonds of these intermediates can be cleaved through ligand substitution reactions to generate macrocyclic structures, [mu-(1,4-(PPh2CH2CH2X)2Y)2(Z)nCu2][PF6]2 (X = S, O; Y = C6H4, C6F4; Z = pyridine, acetonitrile, diimines, isocyanide) in nearly quantitative yields. The incorporation of tetrahedral Cu(I) metal centers into these macrocycles provides a pathway to complexes that differ from analogous d8 square planar macrocycles generated via this approach in their increased air stability, small molecule reactivity, and ability to form multiple structural isomers. Solid-state structures, as determined by single-crystal X-ray diffraction studies, are presented for condensed intermediates and an open macrocycle     

Syntheses and structural characterization of a monomeric tin(II) diamide and a novel chlorotin(II) amide trimer

We have found that addition of a 2:1 ratio of the bulky ligand Li[N(mesityl)(SiMe(3))] to SnCl(2) yields Sn[N(mesityl)(SiMe(3))](2), which is found by X-ray crystallography to be monomeric in the solid state. Interestingly, the solid state structure of the stannylene exhibits a "minipocket" caused by the parallel arrangement of the phenyl rings. A 1:1 ratio of ligand to SnCl(2) affords the new chlorotin amide [Sn(mu-Cl)[N(mesityl)(SiMe(3))]](3), which adopts a unique trimeric structure in the solid state. The chairlike (Sn-Cl)(3) backbone shown here has not been seen previously in Sn-halide chemistry.     

Synthesis and magnetic properties of wheel-shaped [Mn12] and [Fe6] complexes

The reactions of the ligands N-methyldiethanol amine and N-ethyldiethanol amine (abbreviated H(2)mdea and H(2)edea, respectively) with [Mn(12)O(12)(O(2)CCH(3))(16)(H(2)O)(4)] yield novel dodecanuclear wheel-shaped products. The capability of the ligands H(2)mdea and H(2)edea to support wheel structures in metals other than Mn is demonstrated with the crystal structure of a new hexanuclear ferric wheel.