The G[s]G dinucleoside 6 and the G[s]G* dinucleoside 8 were prepared by alkylation of the guanosine thiols derived from 2 and 5, respectively, by the C(8)-chloromethylated guanosine 4 that was obtained from alcohol 3. Dinucleosides 6 and 8 were deacylated to 7 and 9, and fully deprotected to 10 and 11, respectively. The G[N]G dinucleoside 16 was obtained by reductive amination of aldehyde 13 with an iminophosphorane derived from azide 14 and deprotection of the resulting dimer 15. In the solid state of 6, and in a solution of 6 and 8 in CDCl3, H-N(1/I) and H-N(1/II) are engaged in intramolecular H-bonds to the C=O of the isobutyryl protecting groups, and HN of the isobutyryl group of unit I forms an interresidue, intramolecular H-bond to N(7/II), leading to a syn orientation of the nucleobase at unit I, to a tg orientation of the sulfanyl moiety, and to an orthogonal orientation of the nucleobases, preventing any base pairing. The silylated and isopropylidenated dinucleosides 7 and 9 are present in DMSO solution as solvated monoplexes. Broad H-1-NMR signals of the nucleosides 7 and 16 in CHCl3 solution evidence equilibrating G-quadruplexes. The quadruplex formation of 7 and 16 was established by H-1-NMR spectroscopy (only of 16), vapour pressure osmometry, mass spectrometry, and CD spectroscopy. The C(6(I))-hydroxymethylated analogue 9 in CDCl3 and the fully deprotected dinucleosides 10 and 11 in H2O form only weakly pi-pi stacked associates, but no G-quadruplexes, as evidenced by CD spectroscopy.

We investigate the (reduced) Keller-Segel equations modeling chemotaxis of bio-organisms. We present a formal derivation and partial rigorous results of the blowup dynamics of solution of these equations describing the chemotactic aggregation of the organisms. Our results are confirmed by numerical simulations and the formula we derive coincides with the formula of Herrero and Velázquez for specially constructed solutions.

We study the Hamiltonian equations of motion of a heavy tracer particle interacting with a dense weakly interacting Bose-Einstein condensate in the classical (mean-field) limit. Solutions describing ballistic subsonic motion of the particle through the condensate are constructed. We establish asymptotic stability of ballistic subsonic motion.

In the present paper we consider the motion of a very heavy tracer particle in a medium of a very dense, non-interacting Bose gas. We prove that, in a certain mean-field limit, the tracer particle will be decelerated and come to rest somewhere in the medium. Friction is caused by emission of Cerenkov radiation of gapless modes into the gas. Mathematically, a system of semilinear integro-differential equations, introduced in Fröhlich et al. [“Some hamiltonian models of friction,” J. Math. Phys. 52(8), 083508 (2011)10.1063/1.3619799], describing a tracer particle in a dispersive medium is investigated, and decay properties of the solution are proven. This work is an extension of Fröhlich et al. [“Friction in a model of hamiltonian dynamics,” Commun. Math. Phys. 315(2), 401–444 (2012)10.1007/s00220-012-1564-2]; it is an extension because no weak coupling limit for the interaction between tracer particle and medium is assumed. The technical methods used are dispersive estimates and a contraction principle.

We establish the asymptotics of blowup for nonlinear heat equations with superlinear power nonlinearities in arbitrary dimensions and we estimate the remainders.

The phenomenon of Anderson localization is studied for a class of one-particle Schrödinger operators with random Zeeman interactions. These operators arise as follows: Static spins are placed randomly on the sites of a simple cubic lattice according to a site percolation process with density x and coupled to one another ferromagnetically. Scattering of an electron in a conduction band at these spins is described by a random Zeeman interaction term that originates from indirect exchange. It is shown rigorously that, for positive values of x below the percolation threshold, the spectrum of the one-electron Schrödinger operator near the band edges is dense pure-point, and the corresponding eigenfunctions are exponentially localized. Localization near the band edges persists in a weak external magnetic field, H , but disappears gradually, as H is increased. Our results lead us to predict the phenomenon of colossal (negative) magnetoresistance and the existence of a Mott transition, as H and/or x are increased. Our analysis is motivated directly by experimental results concerning the magnetic alloy Eu x Ca 1− x B 6 .

The formation of cyclic duplexes (pairing) of known oxymethylene-linked self-complementary U*[o]A(()*()) dinucleosides contrasts with the absence of pairing of the ethylene-linked U*[c,]A(()*()) analogues. The origin of this difference, and the expected association of U*[x]A(()*()) and A*[x]U-(*()) dinucleosides with x=CH2, O, or S was analysed. According to this analysis, pairing occurs via constitutionally isomeric Watson-Crick, reverse Watson-Crick, Hoogsteen, or reverse Hoogsteen H-bonded linear duplexes. Each one of them may give rise to three diastereoisomeric cyclic duplexes, and each one of them can adopt three main conformations. The relative stability of all conformers with x = CH2, O, or S were analysed. U*[x]A(()*()) dinucleosides with x = CH2 do not form stable cyclic duplexes, dinucleosides with x = O may form cyclic duplexes with a gg-conformation about the C(4')-C(5') bond, and dinucleosides with x = S may form cyclic duplexes with a gt-conformation about this bond.The temperature dependence of the chemical shift of H-N(3) of the self-complementary, oxymethylene-linked U*[o]A(()*()) dinucleosides 1-6 in CDCl3 in the concentration range of 0.4-50 mm evidences equilibria between the monoplex, mainly linear duplexes, and higher associates for 3, between the monoplex and cyclic duplexes for 6, and between the monoplex, linear, and cyclic duplexes as well as higher associates for 1, 2, 4, and 5.The self-complementary, thiomethylene-linked U*[s]A(()*()) dinucleosides 27-32 and the sequence isomeric A*[s]U-(*()) analogues 33-38 were prepared by S-alkylation of the 6-(mesyloxymethyl)uridine 12 and the 8-(bromomethyl)adenosine 22. The required thiolates were prepared in situ from the C(5')acetylthio derivatives 9, 15, 19, and 25. The association in CHCl3 of the thiomethylene-linked dinucleoside analogues was studied by H-1-NMR and CD spectroscopy, and by vapour-pressure osmometric determination of the apparent molecular mass. The U*[s]A(()*()) alcohols 28, 30, and 31 form cyclic duplexes connected by Watson-Crick H-bonds, while the fully protected dimers 27 and 29 form mainly linear duplexes and higher associates. The diol 32 forms mainly cyclic duplexes in solution and corrugated ribbons in the solid state. The nucleobases of crystalline 32 form reverse Hoogsteen H-bonds, and the resulting ribbons are cross-linked by H-bonds between HOCH2-C(8/1) and N(3/1). Among the A*[s]U-(*()) dimers, only the C(8/1)-hydroxymethylated 37 forms (mainly) a cyclic duplex, characterized by reverse Hoogsteen base pairing. The dimers 34-36 form mainly linear duplexes and higher associates. Dimers 34 and particularly 38 gelate CHCl3. Temperature-dependent CD spectra of 28, 30, 31, and 37 evidence pi-stacking in the cyclic duplexes. Base stacking in the particularly strongly associating diol 32 in CHCl3 solution is evidenced by a melting temperature of ca. 2 degrees.