Boundary relations and their Weyl families

The concepts of boundary relations and the corresponding Weyl families are introduced. Let be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space , let be an auxiliary Hilbert space, let

and let be defined analogously. A unitary relation from the Krein space to the Krein space is called a boundary relation for the adjoint if . The corresponding Weyl family is defined as the family of images of the defect subspaces , , under . Here need not be surjective and is even allowed to be multi-valued. While this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space and the class of unitary relations , it also generalizes the notion of so-called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems. Moreover, these new notions yield, for instance, the following realization theorem: every -valued maximal dissipative (for ) holomorphic family of linear relations is the Weyl family of a boundary relation, which is unique up to unitary equivalence if certain minimality conditions are satisfied. Further connections between analytic and spectral theoretical properties of Weyl families and geometric properties of boundary relations are investigated, and some applications are given.

     

Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions. Disordered Phase

The six-vertex model, or the square ice model, with domain wall boundary conditions (DWBC) has been introduced and solved for finite N by Korepin and Izergin. The solution is based on the Yang–Baxter equations and it represents the free energy in terms of an N × N Hankel determinant. Paul Zinn–Justin observed that the Izergin– Korepin formula can be re-expressed in terms of the partition function of a random matrix model with a nonpolynomial interaction. We use this observation to obtain the large N asymptotics of the six-vertex model with DWBC in the disordered phase. The solution is based on the Riemann–Hilbert approach and the Deift–Zhou nonlinear steepest descent method. As was noticed by Kuperberg, the problem of enumeration of alternating sign matrices (the ASM problem) is a special case of the six-vertex model. We compare the obtained exact solution of the six-vertex model with known exact results for the 1, 2, and 3 enumerations of ASMs, and also with the exact solution on the so-called free fermion line. We prove the conjecture of Zinn–Justin that the partition function of the six-vertex model with DWBC has the asymptotics, as N → ∞, and we find the exact value of the exponent κ.The first author is supported in part by the National Science Foundation (NSF) Grant DMS-0354962.     

Substituent control of potassium and rubidium uptake by asymmetric calix[4]-thiacalix[4]tubes

Herein we report on the synthesis and ionophore properties of the first asymmetric p-tert-butylcalix[4]-p-R-thiacalix[4]tubes 7a-c (R = t-Bu, H, 1-adamantyl). The target compounds were obtained by the condensation of tosyloxyethoxy-p-tert-butylcalix[4]arene with the corresponding p-R-thiacalix[4]arenes in the presence of K2CO3 in acetonitrile. The complexation with sodium, potassium and rubidium iodides was studied in CDCl3-CD3OD (4:1) medium by means of 1H NMR measurements. It was found that the ionophore properties of calixtubes 7a-c are controlled by the character of the substituents at the upper rim of the thiacalix[4]arene fragment and it was shown that only the molecular tube 7c with an adamantane-containing thiacalixarene unit is capable of quantitatively binding potassium (swiftly) and rubidium (slowly) cations.     

Microwave-assisted three-component synthesis of 7-aryl-2-alkylthio-4,7-dihydro-1,2,4-triazolo[1,5-a]-pyrimidine-6-carboxamides and their selective reduction

Multicomponent reactions (MCRs) and microwave-assisted organic synthesis (MAOS) have been used as key methods for the synthesis of fused dihydropyrimidine derivatives. The three-component condensation of 3-amino-5-alkylthio-1,2,4-triazoles with aromatic aldehydes and acetoacetamides under microwave irradiation was developed as a rapid and efficient solution-phase method for the high-yielding preparation of 7-aryl-2-alkylthio-4,7-dihydro-1,2,4-triazolo[1,5-a]pyrimidine-6-carboxamide libraries. In addition, the selective reduction of the formed dihydrotriazolopyrimidines to trans-trans-2-alkylthio-7-aryl-4,5,6,7-tetrahydro-1,2,4-triazolo[1,5-a]pyrimidine-6-carboxamides was established. The described synthetic protocols provide rapid access to novel and diversely substituted dihydroazolopyrimidine libraries.     

Inter- and intraconfigurational transitions of Nd3+ in hexafluoroelpasolite lattices

Excitation of the 4f3 ion Nd3+ in hexafluoroelpasolite lattices by synchrotron radiation of wavelength approximately 185 nm leads to fast 4f(2)5d --> 4f3 emission below 52,630 cm(-1) and slower 4f3 --> 4f3 emission from the luminescent states (4)F(3/2) gamma8u (11 524 cm(-1)) and 2G2(9/2) gamma8u (approximately 47,500 cm(-1)). The near-infrared emission is well-resolved, and a clear interpretation of the 4I(9/2) crystal field levels and of the one-phonon vibronic sideband is given. The excitation spectrum of the 2G2(9/2) emission enables clarification of the structure of the 4f(2)5d configuration (which extends from approximately 52,000 to 128,000 cm(-1)). Detailed energy level and intensity calculations have been performed, which provide simulations of the d-f emission and the f-d excitation spectra in good agreement with experiment. It is interesting that although the 4f3 2G2(9/2) gamma8u --> 4f3 4I(J) transitions are very weak in intensity compared with transitions terminating upon higher multiplet terms, most of the 4f(2)5d (3H) 4I(9/2) gamma8g --> 4f3 emission intensity resides in the transitions to 4I(J).     

Modeling of supramolecular ordering and physicochemical properties in cyclohexane-ethanol mixtures

The quasichemical model of the nonideal associated solution (QCNAS) was used to analyze the excess thermodynamic functions and permittivity of cyclohexane-ethanol mixtures in the entire range of compositions and in a wide temperature range. A new model of supramolecular organization was substantiated, taking into account chain and cyclic aggregation of alcohol with the formation of chain aggregates of arbitrary size, which describes a set of physicochemical properties of this mixture with complicated behavior of both thermodynamic and dielectric properties. Equilibrium constants and thermodynamic parameters of H bonding having cooperative character and integral and differential characteristics of aggregation and the structural parameters of aggregates were determined. It was established that the role of nonpolar cyclic structures increases as alcohol concentration decreases. The model describes the permittivity of solutions in the entire range of compositions and reproduces a minimum of the dipolar correlation factor in the range of dilute alcohol solutions. The new data obtained on the supramolecular ordering characterized by the long-range molecular correlations due to H bonding are discussed in detail.     

Absence of long-range magnetic order in the La1.4Sr0.8Ca0.8Mn2O7 bilayered manganite

This paper presents the results of a high-resolution neutron diffraction and magnetometry investigation on the optimally doped (x = 0.3) La(1.4)Sr(0.8)Ca(0.8)Mn(2)O(7) bilayered manganite. In particular, two samples with different oxygen contents have been studied to put in prominence the role of the Mn average valence states at fixed cation concentration. The results show, for the first time, the absence of long-range magnetic order in this optimally doped manganite when the A-site of the structure is doped with equal proportions of isovalent Ca and Sr. This holds for both samples, which present different lattice effects with T, thus suggesting the primary role of cation disorder as the source of the lack of long-range order. The presence, for both samples, of defined insulating- to metallic-like transitions suggests that the transport properties are not linked to the evolution of long-range order and that two-dimensional spin ordering in the layers of the perovskite blocks may be sufficient to "assist" the hole hopping. A possible reason for the suppression of magnetic order induced by the Ca doping is a size effect coupled to the cation size mismatch between the Sr and Ca ions.     

Cryogenic photolysis of activated bleomycin to ferric bleomycin

Activated bleomycin (ABLM) is a drug--Fe(III)-hydroperoxide complex kinetically competent in DNA attack (via H4' abstraction). This intermediate is relatively stable, but its spontaneous conversion to ferric bleomycin (Fe(III).BLM) is poorly characterized because no observable intermediate product accumulates. Light was shown to trigger ABLM attack on DNA in liquid at -30 degrees C, so ABLM was irradiated (at its 350 nm ligand-to-metal charge-transfer transition) at 77 K to stabilize possible intermediates. ABLM photolysis (quantum yield, Phi = 0.005) generates two kinds of product: Fe(III).BLM (with no detectable intermediate) and one or more minor (1-2%) radical O-Fe-BLM byproduct, photostable at 77 K. Adding DNA, even without its target H4', increases the quantum yield of ABLM conversion >10-fold while suppressing the observed radical yield. Since cryogenic solid-phase reactions can entail only constrained local rearrangement, the reaction(s) converting ABLM to Fe(III).BLM must be similarly constrained.     

Driving defect modes of Bose-Einstein condensates in optical lattices

We present an approximate analytical theory and direct numerical computation of defect modes of a Bose-Einstein condensate loaded in an optical lattice and subject to an additional localized (defect) potential. Some of the modes are found to be remarkably stable and can be driven along the lattice by means of a defect moving following a steplike function defined by the period of Josephson oscillations and the macroscopic stability of the atoms.